The Government says it cannot include children at the homes in the scheme
The Coalition of Mother And Baby Home Survivors (CMABS) is demanding to meet the Taoiseach Enda Kenny immediately.
It follows the release of the second interim report of the commission on the issue.
The Mother and Baby Homes Commission of Investigation, chaired by Judge Yvonne Murphy, is in the process of investigarting 14 named homes and a representative sample of county homes.
It has recommended the Government reconsider including children who lived without their mothers at such homes in the State redress scheme.
The report reads: "The Commission considers that the exclusion of most of the Mother and Baby Homes and all of the County Homes from the redress scheme warrants further investigation.
"It considers that the Department of Education and Skills and the Department of Health should re-examine the decisions not to include them".
It continues: "The Commission recognises that any decision about redress for any particular group is a matter for the Government but it is clear that children who were resident in the named Mother and Baby Homes and all of the County Homes without their mothers have a real cause for grievance".
However, the Government say it is not appropriate to do that at the moment.
CMABS say the decision to exclude "an elderly and dying survivor community" is "immoral, repulsive and cold hearted".
The group say the refusal is "kicking the can down the road while waiting for more and more elderly survivors to die".
"Clearly money means more to Zappone and this Government than any sense of common decency or morality", it adds.
It has also criticised the Governments for "hiding" the report for several months.
A stamp on the report shows it was received by the Department of Children and Youth Affairs on September 16, 2016 - but it was only published on Tuesday.
Spokesman for the survivors group, Paul Redmond, told Newstalk Breakfast they want to meet the Taoiseach - and are also hoping to meet Children's Minister Katherine Zappone.
"The religious orders should be responsible for this redress - they've been let off the hook in the past, and they should be firmly nailed on to it this time.
"They still own extensive lands, property and massive cash reserves in this country.
"So I don't see why the Government is afraid if paying redress, when they should really be going after the religious orders".
"I put in for another meeting with Minister Zappone immediately because she did actually invite groups to meet her.
"It's not going to be an easy meeting, I can tell you that for a start.
"We're going to be here to the end and we're going to be here way past the end until we get full inclusion".