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More than a quarter of the women and girls subjected to harsh discipline and unpaid work at the 10 laundries, run by Catholic nuns, were sent there by the Irish state, an official report that ran to almost 1,000 pages said on Tuesday (local time).
The laundries have been accused of treating inmates like "slaves" for decades of the 20th century, imposing a regime of fear and prayer on girls sometimes put in their care for simply falling pregnant outside wedlock. One in 10 inmates died in care, the youngest at 15.
Kenny said on Wednesday that he was sorry for the women who had to live in such conditions but again stopped short of a full state apology, further angering groups representing women who were housed in laundries as recently as 1996.
He said the government needed time to consider the report before it was debated in parliament in two weeks' time - but members of Kenny's junior coalition partner Labour, including a junior minister, said an apology was needed far sooner.
"The government will make that decision and I don't sit around the cabinet table, but my personal opinion is that there should be an apology," Kathleen Lynch, junior minister with responsibility for the elderly, told Today FM radio.
"The stigma that has been attached to these women, the blight it has been on their lives, there's nothing to be proud of here. You can say that everyone suffered in the 50s, yeah, but some suffered more than others."
Lynch was referring to Kenny's comments in parliament on Tuesday when he said the laundries, described in the report as lonely and frightening places, had to be seen in the context of an Ireland that was an uncompromising and authoritarian place.
"STUFF YOUR APOLOGY"
Criticism of Kenny also dominated the front pages of Wednesday's newspapers alongside details from the report. The Irish Sun's headline read "Stuff Your Apology", while the Irish Daily Mail simply asked "Why Can't You Say Sorry?"
Irish governments had in the past denied blame, emphasising the laundries were private institutions, but the report concluded there was "significant state involvement".
The matter of an apology was raised by Labour lawmakers at the parliamentary party's weekly meeting on Wednesday. One Labour member of parliament, Dominic Hannigan, said the government needed to be seen to take responsibility.
"I think it's time for an apology, people have been waiting a long, long time for this," he told national broadcaster RTE.
Labour leader and Deputy Prime Minister Eamon Gilmore is likely to come under pressure to apologise in parliament when he takes leaders' questions on Thursday, with opposition parties also calling on the government to meet survivors' appeals for compensation for their treatment in the laundries.
Unlike other harrowing reports where priests were found to have beaten and raped children in Catholic-run institutions, no allegations of sexual or physical abuse were made against the nuns at the laundries, Tuesday's report said.
But former inmates spoke of an uncompromising regime of physically demanding work and prayer enforced by scoldings and humiliations.
The report's findings follow investigations into clerical sex abuse and state-abetted cover-ups that have shattered the authority of the church in Ireland and rocked the Catholic Church's reputation worldwide.