On invariant (co)homology of a group

There are different notions of homology and cohomology that can be defined for a group with an action of another group by group automorphisms. In this paper we address three natural questions that arise in this context. Namely, the relation of these notions with the usual (co)homology of a semidirect product, the interpretation of the first homology group as some kind of abelianization and the classification of (invariant) group extensions.