Labour are going to get around 19 of the 27 select committee chairs because of their seats. Lets assume they will go for the women/equalities committee. The chair has to be from that party and any backbencher can stand.
Chairs are elected by all MPs in a secret ballot with transferable vote.
Details on the current outlook for chairs. It doesn't discuss the Womens committee:
https://archive.is/YPfA8
It is definitely possible that a pro-woman Labour candidate will stand Rosie, Tonia or Marsha de Cordova perhaps. Remember Rosie was elected chair of the women's PLP twice. As much as I love Rosie the strategic move is someone like Tonia or Marsha de Cordova standing who are just as committed, but are less controversial. I imagine the discussions are ongoing right now about the best plan maybe secret cross party discussions. Alternatively they could agree on backing someone pro-women but who has been more on the sidelines.
While a lot of Labour MPs sometimes like to pretend women don't exist in public in a secret ballot their opinion may be different. We already saw that with Rosie getting elected. Also most Tories would definitely give a pro-woman candidate first votes. So getting the chair is definitely a possibility with some cunning moves.
One factor will depend what the Starmer government thinks. For instance they might not like Kate Osborne or those like her as chair as she will be criticising them for not supporting men enough. Also perhaps more importantly Osborne is from the further left so I think she will not have the votes at all. They also might not want someone really pro-woman as she will calling them out from the other side.
So they might push a middle of the road candidate.
However as I said that doesn't mean they would win. Whatever happens it will be a very interesting indication of what MPs really think about women's rights.