Given a group $G$, we have functors $H^n(G; -)$ taking a $G$-module $M$ to its cohomology. One of the most important features of such functors is that short exact sequences of $G$-modules induce a long exact sequence in cohomology: given $0 \to N \to M \to L \to 0$ we have a long exact sequence $$ ... \to H^{n}(G; N) \to H^n(G; M) \to H^n(G; L) \to H^{n-1}(G;N) \to \hspace{4px} ... $$ I see this fact as saying that in essence, group cohomology is actually about modules rather than groups.
This is also because I find it very hard to replicate this behavior for the category of groups instead of those of $G$-modules:
- I couldn't really make good sense of functoriality in the first parameter of $H^n$. The best I could manage is to show that if $\phi : G \to H$ is a morphism of groups, and $M$ is an $H$-module, and we induce a $G$-action on $M$ via $\phi$, than we have a morphism $H^n(H; M) \to H^n(G;M)$. (I'm pretty sure that this claim is true).
- I definitely could not make sense of making a long exact sequence in group cohomology out of a short exact sequence of groups $1 \to N \to G \to H \to 1$.
I'm interested in the question of relating the group cohomology of different groups, even in ways that are not directly related to either 1 or 2. I'd also be happy to hear more about these perspectives being applied to group cohomology.