The House of Commons public accounts committee has announced an investigation into the Home Office’s treatment of thousands of international students who may have been wrongly accused of cheating in an English language test they were required to sit as part of a visa application process.

More than 100 students signed a letter calling on the home secretary, Sajid Javid, to make a long-promised statement on the issue and to act swiftly to resolve the situation in the time he has remaining before a possible government reshuffle once the Conservative leadership campaign ends.

The letter was delivered to the Home Office on Thursday by a group of students who say they have been mistakenly accused of cheating.

About 2,500 students have been forcibly removed from the UK after being accused of cheating in the exam and a further 7,200 left the country after being warned they faced detention and removal if they stayed. Many have protested their innocence: 12,500 appeals have been heard in UK courts, and so far 3,600 people have won their appeals.

“We are some of the tens of thousands of international students unjustly robbed of our visas and our rights by the Home Office in 2014 after we were accused of cheating on an English test,” the students’ letter reads. “We are innocent but the government gave us no real way to defend ourselves, so we’ve been fighting for five years to clear our names. The department you lead ruined our lives and stole our futures. It branded us as frauds, forcing us to bear a lifelong mark of shame, while never presenting any evidence at all against most of us.”

The students added: “Many of us are destitute, barely able to live from day to day. Many of us are on medication for stress or depression. Many of us have been rejected by our families, who are shamed by the allegation of cheating. Some of us have tried to kill ourselves.”

A National Audit Office investigation into the Home Office response to reports of cheating in English language tests concluded last month that some people may have been wrongly accused and unfairly removed from the UK. The public accounts committee has invited the permanent secretary at the Home Office, Sir Philip Rutnam, to answer questions at a hearing into the issue in July.

In 2014, undercover filming by BBC reporters for Panorama revealed organised cheating in two of the 90 Home Office-approved centres offering the test of English for international communication (Toeic) exam, which is required for visa extensions. The government responded by asking the test provider, the US-based Educational Testing Service (ETS), to assess whether the 58,459 tests taken between 2011 and 2014 were valid. The US company made checks and concluded that virtually every test taken was suspicious, identifying 97% of all UK tests as “suspicious”. It classified 58% as “invalid” and 39% as “questionable”.

The Home Office suspended the licences of some test centres and revoked the visas of those accused of cheating. Campaigners have questioned whether it is likely that 97% of people taking Toeic tests could have been involved in cheating.

The Labour MP Stephen Timms was told by Javid last year that he was “sympathetic” to the students’ situation. Timms also urged Javid to use the time remaining to him in his position as home secretary to resolve the issue. An all-party parliamentary group on the Toeic issue, chaired by Timms, heard this month that the evidence sent to the Home Office by ETS was “unsafe and unreliable”, Timms said. “It is urgent that this is addressed for a large number of students who have been suffering a great deal for a very long time,” he added.

Nazek Ramadan, the director of Migrant Voice, a charity that has been supporting the students, said this was a direct plea to the home secretary “who has the power to end this injustice and give them back their future” to make this issue a positive legacy from his time at the Home Office. “Some of them are prone to be detained or deported any time. We know now that the Home Office action was based on unreliable evidence. This should be put right,” she said.

A Home Office spokesperson said 25 people involved in the organised cheating exposed by Panorama had received criminal convictions. “The home secretary is considering the findings of the NAO report. He will then make a statement to parliament.”