Ken Brown shows in
- Homological criteria for finiteness, Comment. Math. Helv. 50 (1975), 129–135, doi:10.1007/BF02565740, (free author version)
that group cohomology for a group G commutes with direct limits iff G is of type $FP_\infty$. That is the trivial module $\mathbb Z$ has a projective $\mathbb ZG$ resolution which is finitely generated in each degree.
