Let $G$ be a profinite topological group, $M$ a discrete $G$-module.

If $M$ is "P", is every $H^i_{\rm cont}(G,M)$ also "P"? or at least is it a subgroup/subquotient of an abelian group that is "P"? for "P" one of the following properties:

- torsion divisible.
- finitely generated as an abelian group.

I would benefit from a reference.