This is much more mundane than the main intended applications of Condensed Mathematics, but still useful. It is often interesting to consider $H^*(G;M)$, where $G$ is a profinite group and $M$ is some kind of topological abelian group with action of $G$. Here $G$ could be the Galois group of an infinite field extension (in algebraic number theory) or the Morava stabiliser group (in chromatic stable homotopy theory). There are various results about vanishing of certain cohomology groups, duality phenomena, special features of the case where $G$ is $p$-adic analytic or uniformly powerful, and the relationship with cohomology of rational Lie algebras. The traditional treatment of this story has many awkward features caused by the fact that topological abelian groups do not form an abelian category. Different theorems have a maze of different technical conditions formulated in different subcategories of coefficient modules, and it is hard to keep everything straight. It should be much cleaner to redevelop the theory using the abelian category of condensed abelian groups. As far as I know, no one has yet written out the details. However, I expect that it should not be too hard, and that a strong student could do it as a masters thesis, for example.