A cross-party group of 56 MPs have called on the culture secretary, Karen Bradley, to explain her decision to block the only female BME applicant to Channel 4’s board.

It emerged this week that the Department for Culture, Media and Sport had taken the unusual step of vetoing the unnamed candidate, who was one of five put forward by the regulator, Ofcom. The remaining four applicants who were accepted are all white men, and the Channel 4 board has no members from ethnic minorities currently sitting.

Though only four roles were originally advertised by Channel 4, Ofcom decided to put forward what it considered to be five “outstanding” candidates. The regulator, Channel 4 and the government have declined to identify the rejected candidate, but one source said she was not a well known public figure.

In a letter to Bradley, the MPs describe the decision to reject a rigorously vetted person at such a late stage as unprecedented, and say it is a “matter of urgency to find out the reasoning behind your decision to reject the candidate”.

Among the signatories was David Lammy, a former culture minister, who said the government had “serious questions to answer”.

“The government talks a good game on gender and racial diversity in the boardroom but actions speak louder than words,” said Lammy, who is also chair of the all-party parliamentary group on race and community. “Channel 4’s remit requires it to reflect the diversity of the UK and the organisation is making great progress in this regard.

“In appointing four white men to the board, the secretary of state has failed Channel 4 and the country, and in doing so has hugely undermined all of the work being done on diversity across the broadcasting industry.”

Responding to the letter late on Friday, Bradley said the department had picked the “best four candidates” for the four roles advertised by Channel 4, but appeared to leave open the possibility of appointing other board members with other skill sets at a later date.

Bradley’s letter read: “I am very grateful for your work to support diversity, particularly in our creative industries. I am in full agreement in the need to ensure our public institutions and appointments represent and reflect modern Britain. I do however firmly reject any suggestion that female or BAME candidates are treated unfairly in our public appointment decisions.”

Her comments echoed those of Theresa May, who hen challenged over Bradley’s decision by Lammy at prime minister’s questions, defended the government’s appointments process. She said she was not aware of the specific case but it was “about who is right for the job”.

The shadow culture secretary, Tom Watson, shadow education secretary, Angela Rayner, shadow minister for diverse communities, Dawn Butler, shadow home secretary, Diane Abbott, Conservative former minister Helen Grant and SNP spokesman for culture, media and sport, John Nicolson, were also among those who signed the letter.

Butler said May and the government would be judged on what they said and did on improving diversity at the highest levels.

“Change does not come about through talk but through action,” she said. “The five candidates went through a rigorous vetting process and were of an extremely high calibre. This is an unprecedented move by this Tory government as it is the first time that a candidate has been rejected.

“To not select the only woman and person of colour is extremely worrying and the government should be held to account for this decision.”