Reacting to the publication of Ms Judge Laffoy's resignation letter, Mr Colm O'Gorman of One in Four said that, while the letter was highly critical of the Government, "the focus shouldn't be put on blame and recrimination".
"What we need to do is establish quickly what went wrong," he said. "The only way to do that is through an independent Oireachtas inquiry."
"I think Justice Laffoy's letter makes it absolutely imperative that this be done as soon as is possible."
He believed the contents of the Laffoy resignation letter, published in the Sunday Tribune yesterday, "undermines completely and absolutely the integrity of the Government on this issue".
The letter accused the Government of failure to properly resource and support the inquiry. "That is why we need an independent and transparent inquiry," he said.
Mr O'Gorman believed the inquiry was necessary to ensure any future commission of inquiry does not encounter the same problems as Laffoy.
Ms Christine Buckley accused the Government of deliberately obstructing the Laffoy Commission.
"The Government did not want the commission to continue," she said.
Ms Buckley believed the Government was instead relying "as a very easy option" on the Redress Board compensation scheme as the primary way of dealing with the issue.
She also called on the Government to implement the reforms proposed by Ms Justice Laffoy last year, as opposed to carrying out a second review.
"I would still be desperately hoping that Mary Laffoy will reconsider her resignation but I could not see her doing so under the current Minister for Education."
Survivor group SOCA Ireland, which had opposed the inquiry over its structure, said it felt vindication about its non co-operation with the Laffoy Commission.
Reacting to the letter, Mr John Kelly said the inquiry had been dealt "a mortal blow and should be pronounced dead and buried".
He said that "transparency and openness are a prerequisite for those whose trust has been violated once again".
"The bona fides of the victims [in relation to the inquiry] have been questioned. It should have been the other way round. A lot of people have been conned," he added.