I started learning about group cohomology (of finite groups) from two books: Babakhanian and Hilton&Stammbach. The theory is indeed natural and beautiful, but I could not find many examples to its uses in algebra.
I am looking for problems stated in more classical algebraic terms which are solved elegantly or best understood through the notion of group cohomology. What I would like to know the most is "what can we learn about a finite group $G$ by looking at its cohomology groups relative to various $G$-modules?").
The one example I did find is $H^2(G,M)$ classifying extensions of $M$ by $G$.
So, my question is:
What problems on groups/rings/fields/modules/associative algebras/Lie algebras are solved or best understood through group cohomology?
Examples in algebraic number theory are also welcome (this is slightly less interesting from my current perspective, but I do remember the lecturer mentioning this concept in a basic algnt course I've taken some time ago).