In May this year, a huge company listed on the London Stock Exchange found itself in the midst of controversy about a prison it runs for the government – Thameside, a newly built jail next to Belmarsh, in south-east London. A report by Her Majesty's Inspectorate found that 60% of its inmates were locked up all day, and there were only "vague plans to restore the prison to normality". The prison campaign group the Howard League for Penal Reform talked about conditions that were "truly alarming".

Two months later, the same company was the subject of a high- profile report published by the House Of Commons public accounts committee, prompted by the work of Guardian journalist Felicity Lawrence. This time, attention was focused on how it was managing out-of-hours GP services in Cornwall, and massive failings that had first surfaced two years before. Again, the verdict was damning: data had been falsified, national standards had not been met, there was a culture of "lying and cheating", and the service offered to the public was simply "not good enough".

Three weeks ago, there came grimmer news. Thanks to its contracts for tagging offenders, the company was now the focus of panic at the Ministry of Justice, where it had been discovered that it was one of two contractors that had somehow overcharged the government for its services, possibly by as much as £50m; there were suggestions that one in six of the tags that the state had paid for did not actually exist. How this happened is still unclear, but justice secretary Chris Grayling has said the allegations represent something "wholly indefensible and unacceptable".

The firm that links these three stories together is Serco. Its range of activities, here and abroad, is truly mind-boggling, taking in no end of things that were once done by the state, but are now outsourced to private companies. Amazingly, its contracts with government are subject to what's known as "commercial confidentiality" and as a private firm it's not open to Freedom of Information requests, so looking into the details of what it does is fraught with difficulty.

But the basic facts are plain enough. As well as five British prisons and the tags attached to over 8,000 English and Welsh offenders, Serco sees to two immigration removal centres, at Colnbrook near Heathrow, and Yarl's Wood in Bedfordshire. You'll also see its logo on the Docklands Light Railway and Woolwich ferry, and is a partner in both Liverpool's Merseyrail network, and the Northern Rail franchise, which sees to trains that run in a huge area between the North Midlands and English-Scottish border.

Serco runs school inspections in parts of England, speed cameras all over the UK, and the National Nuclear Laboratory, based at the Sellafield site in Cumbria. It also holds the contracts for the management of the UK's ballistic missile early warning system on the Yorkshire moors, the running of the Manchester Aquatics Centre, and London's "Boris bikes".

As evidenced by the story of how it handled out-of-hours care in Cornwall, it is also an increasingly big player in a health service that is being privatised at speed, in the face of surprisingly little public opposition: among its array of NHS contracts is a new role seeing to "community health services" in Suffolk, which involves 1,030 employees. The company is also set to bid for an even bigger healthcare contract in Cambridgeshire and Peterborough: the NHS's single-biggest privatisation – or, if you prefer, "outsourcing" – to date, which could be worth over £1bn.

But even this is only a fraction of the story. Among their scores of roles across the planet, Serco is responsible for air traffic control in the United Arab Emirates, parking-meter services in Chicago, driving tests in Ontario, and an immigration detention centre on Christmas Island, run on behalf of those well-known friends of overseas visitors the Australian government.

In the US, the company has just been awarded a controversial $1.25bn contract by that country's Department of Health. All told, its operations suggest some real-life version of the fantastical mega-corporations that have long been invented by fiction writers; a more benign version of the Tyrell Corporation from Blade Runner, say, or one of those creations from James Bond movies whose name always seems to end with the word "industries".

The strangest thing, though, is the gap between Serco's size and how little the public knows about it. Not for nothing does so much coverage of its work include the sentence "the biggest company you've never heard of".

I first heard Serco's name about eight years ago, when I was just starting to understand the amazing growth of what are now called "public service companies". Once I started looking, their logos were everywhere, suggesting a shadow state that has since grown ever-bigger. Their names seemed anonymously stylised, in keeping with the sense that they seemed both omnipresent, and barely known: Interserve, Sodexo, Capita, the Compass Group.

A Serco clinic in Truro, Cornwall. Photograph: Jim Wileman

Serco is among the biggest of them all. At the last count, its annual pre-tax profits were up 27%, at £302m. In 2012 alone, its British workforce grew by 10,000, to 53,000 people (tellingly, as many as 90% of them are said to be former civil servant employees). In terms of employees, that makes it more than twice as large as the BBC, and around 20% bigger than Philip Green's Arcadia group. A very significant player, in other words, and one that has come a long way since its foundation 1929, when it was a branch of the American RCA corporation called RCA Services Ltd, involved in the then booming UK cinema industry. It was renamed Serco in 1987, after a management buy-out, and floated on the stock exchange the following year. In the 25 subsequent years, during which the UK has grown ever-fouder of outsourcing and privatisation, Serco has grown at an amazing rate.

The current chief executive of the global Serco Group is 49-year-old Chris Hyman, born in Durban, South Africa. His annual remuneration is around £700,000, plus bonuses; in 2011, the value of his total package rose 18%, to £1.86m (the company's finance director had to slum it at £948,295).

In 2010, Hyman was given a CBE for services to business and charity; he is also an enthusiastic fan of motor racing and an evangelical Christian. Four years ago, he was asked about his company's very low profile, and he said this: "We had a dilemma – what do we do with the Serco name. We are proud of it. We thought we needed billboards at airports and places like that, to be seen with Tiger Woods on. But we worked out very quickly that is not what we are meant to do. We are meant to be known by the 5,000 not the five billion. The people who serve the people need to choose who supplies the service. We are delighted when the public knows who we are, but really, we need to be known by the people who make decisions."

When Serco made its bid to run NHS community-health services in Suffolk – district nursing, physiotherapy, OT, end-of-life palliative care, wheelchair services – it reckoned it could do it for £140m over three years – £16m less than the existing NHS "provider" had managed, which would eventually allow for their standard profit margin of around 6% a year. When it started to become clear that Serco was the frontrunner, there was some opposition, but perhaps not nearly enough. "Suffolk isn't the most politically active part of the country," says one local insider. "And the staff were very lackadaisical. It was: 'NHS Suffolk wouldn't made a bad decision.' So it was hard to get a campaign going."

Serco was officially awarded the contract in October 2012, which meant that hundreds of staff would leave the NHS, and become company employees. Within weeks, the company proposed a huge reorganisation, which involved getting rid of one in six jobs. This has since come down to one in seven, two thirds of which will apparently go via natural wastage. In terms of their pay and conditions, the hundreds of people who have been transferred from the NHS to Serco are protected by provisions laid down by the last government, but it is already becoming clear that many new staff are on inferior contracts: as one local source puts it, "they've got less annual leave, less sick pay … it's significantly worse."

Meanwhile, other people are reportedly quitting their jobs, and the service given to patients is said to be getting worse. "In my team alone, we're 50% down on staffing hours compared with last year," says one former NHS worker, who provides home-care to patients who are largely elderly. Thanks to poor morale, she says that the team in which she works has lost around a third of its staff, and she is also having to see to administrative tasks that were previously carried out by someone else: in addition, she claims, support for a new IT regime is "farcical".

"We've still got the same number of patients," she says, "so the workload has massively increased." As a result, she and her colleagues are having to cut people out of their previous entitlement to treatment at home. "That completely goes against our ethics," she says, "but that's what we're having to do."

Phosphate Hill Detention Centre in Australia. Photograph: Getty Images

The NHS is a relatively new area of controversy for Serco, but concerns about their practices run across many other areas. Right now, the controversy over alleged overcharging, focused on both Serco and its fellow tagging- contractor G4S, seems to have only just begun. When the news was made public, 8% was wiped off Serco's share price. The Cabinet Office has announced a review "into government-held G4S and Serco contracts to ensure that contracts are well-managed and in good order", which will report in the autumn. Work for the British government accounts for 40% of Serco's revenues; to quote from the Daily Telegraph. "Without Serco, Britain would struggle to go to war". That gives you some idea of how deeply its work penetrates the state, and how unthinkable any kind of corporate crisis would be.

Margaret Hodge, the former Labour minister who now chairs the public accounts committee, clearly thinks that all these stories point to huge issues. She talks about "the inability of government to contract-out in a way that protects the taxpayer's interest." The Cornwall out-of-hours story, she tells me, was reducible to "an absurd situation where you had a company seemingly lying about what it was doing, but there was nothing in the contract that could allow you to terminate it – indeed, they still appeared to be eligible for their bonus payments. It's quite extraordinary."

There are even bigger issues at stake, though. "There's also the inability of the public sector to monitor effectively," she says. "The Cornwall story came to light because of a Guardian journalist and a whole load of whistleblowers. Which is nuts: a crazy way for the public sector holding to account the private sector when it's delivering public services." Even her committee, she says, cannot break through a great wall of commercial confidentiality, and look at what the companies delivering pubic services are up to – not just in terms of their bids for public services and contracts with government, but such vital matters as their costs, and the profits they make from particular jobs.

Does she feel any guilt about the fact that companies such as Serco made their decisive breakthrough into public services when Labour was in power? There's a murmur of agreement. "I think we were as bad at managing this diversity of providers," she says. "But one of the things that gets me with this government is that they should have learned from our mistakes. What is becoming really clear to me … is that the Sercos, the A4s, the G4Ss, the Capitas – they're good at winning contracts, but too often, they're bad at running services."

And what of the incredible range of what Serco actually does, from school inspections to Boris bikes? "Interestingly, we are looking at this. The National Audit Office is doing work around the development of quasi-monopoly private providers, which is the world we're moving into. We don't really understand the size of their empires. We've got to start getting hold of this. It's a new phenomenon."

Once I'd spoken to Hodge, I got hold of one of Serco's "media relations team", and arranged to send him a few questions. On the subject of the out-of-hours GP fiasco in Cornwall, he quoted a response from the doctor in charge of their set-up. "It's really important that the local people in Cornwall do not lose confidence in this essential urgent care service," he said. "It is a valued part of the local NHS and we are proud of our professional team who provide it." A wider statement said the company had taken "swift and decisive action to put the situation right and apologised to the people of Cornwall", and made "a goodwill gesture to repay the bonus made [sic] to us in 2012, which we were under no obligation to do." All told, I was assured, their service "delivers a high standard against the national quality requirements".

On the allegations about what has happened since Serco took over community healthcare in Suffolk, and the claim that any new starters aside from clinical staff are on inferior terms and conditions, the same spokesperson said that such employees are "offered contracts in line with Serco standard terms and conditions which are market comparable". He denied that anyone had been cut out of treatment at home, said that the company had "recently realigned our clinical teams across Suffolk according to the needs of the areas in which they deliver care" and claimed that new IT systems are being implemented "slowly and carefully".

The controversy surrounding Thameside prison, they said, had been followed by "a series of initiatives" including a "gangs strategy", and measures to help prisoners with mental-health issues. Some people were now allowed to be outside their cells "during the core part of the day", and in August, Serco anticipated that this would be extended.

As for the ongoing story about overcharging for their tagging contract with the Ministry of Justice, Serco said this: "We are working with our customer, the Ministry of Justice [on] this matter so there is very little we can add at this stage." I was also directed to a statement from Chris Hyman, which said the company "will not tolerate poor practice and behaviour and wherever it is found we will put it right", and reminded that justice secretary Chris Grayling has said he so far has "no information to confirm dishonesty had taken place" on the part of either Serco or G4S.

There was one last question, concerning the amazing spread of what Serco sees to, from parking meters, through nuclear early warning systems, to an expanding share of the NHS. Is there any limit to the fields they work in?

"We operate in a range of markets and geographies," went its answer, "which means we are well placed to bring a wide range of experiences and knowledge to help customers with the challenges that they face."

That'll be a no, then.