There have been calls for the Titus Trust to close over its handling of abuse perpetrated by the late John Smyth QC.
The trust on Friday said a settlement had been reached with three men "who have suffered for many years because of the appalling abuse" perpetrated by Smyth while chair of the Iwerne Trust - forerunner to the Titus Trust - during the 1970s and 80s.
Smyth died in 2018 after being exposed by Channel 4 as having abused young men he groomed through Iwerne.
In its latest statement, the Titus Trust said: "We are devastated that lives have been blighted by a man who abused a position of trust and influence to inflict appalling behaviour on others, and we have written to those concerned to express our profound regret at what happened and also to apologise for any additional distress that has been caused by the way The Titus Trust has responded to this matter."
It said the details that have emerged in recent years have caused the trust to "reflect deeply on our current culture and the historic influences upon us".
It is inviting feedback from those who have had some involvement with the trust and its camps as part of an upcoming independent cultural review.
"The Trustees regret that we have not been able to speak out while the legal situation has been ongoing and want to take the opportunity now to listen well to people's experiences of our camps to inform our future planning," the trust said in a statement.
An independent review into the Church of England's handling of allegations against Smyth is currently being undertaken by Keith Makin, a former director of social services. The trust said it was "co-operating fully" with the review.
"We are sorry that the Titus Trust's earlier public statements were inadequate as explanations of the relevant facts and history and that some of the language the Trust has used in public statements about these matters has prompted anger on the part of some survivors and others," the statement continued.
"We recognise the impact that this guarded use of language has caused, and apologise if this has contributed in any way to the anguish experienced by the survivors and their families."
However, Smyth's victims, in a statement released through their spokesman Andrew Graystone, have rejected the apology and called for the Titus Trust to immediately cease activities and disband.
The victims accused the trustees of doing "everything they could to protect their own interests" and failing to take responsibility.
"We are pleased that they have finally issued a limited apology for their recent behaviour, but we note that none of those responsible has resigned," they said.
"They have not acknowledged the historic cover-up. There is no evidence that the culture of moral superiority, exclusivity and secrecy that has pervaded the network for decades has changed in any way."
They added: "A culture that has resisted reform in the face of overwhelming evidence of damage over many years is beyond reform. It is our wholehearted belief that in the light of these events the Titus Trust and its work should cease immediately."
The Titus Trust has been contacted for comment in response to the statement from victims.