Almost four fifths of Fine Gael backbenchers who responded to a recent survey support the holding of a referendum on gay marriage.
Two-thirds also want a re-run of the referendum on giving powers of inquiry to the Oireachtas, but just over half support the abolition of the Seanad.
A total of 43 of the 50 Fine Gael backbenchers took part in the survey for Newstalk radio.
Off the back of the vote on abortion, 88% said they would still have supported it had there been a free vote.
Some 79% said that they support the holding of a referendum on full marriage access for gay couples.
A small majority, 56%, agree with full parental and guardianship rights being extended to same-sex couples.
Almost two thirds, or 65%, believe another referendum should be held on giving powers of inquiry to the Oireachtas.
On Abolition of the Seanad, the party is very divided - 54% agreeing with abolition, 46% against it and 2% unsure.
Finally, 67% want a Cabinet reshuffle - opening the possibility of a promotion.