Government ministers were today accused of lying in an attempt to cover up Britain's role in the so-called extraordinary rendition of two men captured in Iraq and sent to Afghanistan for interrogation.

Both men have been detained without trial or charge at a US interrogation centre north of Kabul after British special forces troops who detained them near Baghdad early in 2004 handed them over to US forces.

The legal charity Reprieve today said John Hutton, as defence secretary, had misled parliament when he said the men were members of a banned Sunni Muslim militant organisation, Lashkar-e-Taiba. After identifying one of the men, Amantullah Ali, and contacting his family in Pakistan, Reprieve discovered that he is not Sunni but Shia.

The charity said Hutton was wrong to suggest the two men needed to be moved to Afghanistan because the US had insufficient linguists to interrogate them in Iraq, as at least one of them spoke Arabic, and to claim that they were being held in humane conditions that met international standards.

Reprieve's attempts to represent the two men in legal proceedings were hampered by the Ministry of Defence's refusal to identify them, citing their "data protection rights".

According to Reprieve, Ali is a rice merchant from Pakistan who was on pilgrimage in Iraq when he was shot in the foot and captured. The identity of the second man has not been confirmed but he is thought to be called Salahuddin. Reports from Afghanistan suggested he had suffered "catastrophic" mental health problems during his five-and-a-half years in detention, Reprieve said.

Clive Stafford Smith, the director of Reprieve, said Ali had five children, one of whom was five months old when he was detained. "I hope people in government who have children will consider those five-and-a-half years of a child's life and consider how horrible it must be to miss them."

Reprieve pointed out that Tony Blair and Jack Straw had denied any British involvement in the US rendition programe, with Straw telling the Commons' foreign affairs committee in December 2005: "Unless we all start to believe in conspiracy theories and that the officials are lying, that I am lying, that behind this there is some kind of secret state which is in league with some dark forces in the United States, and also let me say, we believe that Secretary [Condoleezza] Rice is lying, there simply is no truth in the claims that the United Kingdom has been involved in rendition, full stop."

The MoD has since confirmed that British officials became aware of the "intention to transfer" Ali and the second man from Iraq to Afghanistan in March 2004, and by June that year were aware that they had been transferred.

Clive Stafford Smith, the director of Reprieve, said: "Government ministers misled Parliament and the country by denying that we had anything to do with rendition and then, when John Hutton apologised for this, he misled Parliament again. The government now appears unwilling to admit that they are still propagating falsehoods."

Today the MoD continued to insist that both men were members of Lashkar-e-Taiba, despite evidence that one of them is a Shia Muslim. A spokesman said: "Their initial detention was appropriate, legitimate and targeted at saving lives."