The human rights advocacy group at the centre of the "Jihadi John" revelations has come under pressure by MPs for not publicly condemning the terrorist attacks in Paris last weekend.
Cage, an organisation that claims to "empower communities impacted by the War on Terror", was criticised by MPs on the home affairs select committee for not putting out a public statement.
Asked on Tuesday why the group, which has been criticised by some for sharing platforms with "radical extremists", did not put out a statement, director Dr Adnan Siddiqui condemned the attacks but said that Muslims should not feel responsible.
"I think what Muslims have said, after the initial emotion related to it, and no one will say they weren't disgusted by what happened, ... is to say, were we responsible?" Siddiqui said.
He added: "I don't think we have anything to say in respect to it. Our condemnation means little," and said that those wanting to see a public apology from Cage usually had an ulterior motive.
Cage's press officer, Ibrahim Mahmoud, said Cage had been busy on the Friday of the attacks "dealing with emails and media requests" in relation to Mohammed Emwazi aka "Jihadi John", who had reportedly been killed in a drone strike earlier that day.
Cage had been called into parliament to contribute to the committee's ongoing inquiry into how to counter extremism in Britain, particularly looking into how young Britons travelling to fight in Syria had been radicalised.
The committee also questioned the group on claims that some members "supported the idea of an Islamic state" and whether it differed to that of ISIS. Though Siddiqui said he supported the idea of a "caliphate that is modelled similarly to the Ottoman state", he clarified this was his personal view and the question "was not in Cage's remit".
However, members of the committee expressed concern, saying that because of Cage's high profile it was "incumbent on you, and other Muslim organisations to proactively condemn, and say this is an affront" to Islam.
The advocacy group came into the media spotlight earlier this year when reports revealed that Emwazi, who was from west London, had met with members of the organisation prior to joining ISIS.
Some also criticised the group's research director, Asim Qureshi, for sympathising with Emwazi after he called him a "beautiful young man". However, Cage insists the comment was only in reference to Emwazi while he was still residing in the UK.
The committee also met with the heads of Bethnal Green Academy, which three schoolgirls left to travel to Syria in March, and representatives from the activist organisation Inspire, who work on antiextremism programmes in schools.