I am reading about Hilbert's Theorem 90 in James Milne's excellent notes about Galois theory. The author introduced the notion of a $G$-module, which is just an abelian group together with a group homomorphism $G\to \text{Aut}(M)$. The author himself didn't mention the category of $G$-modules explicitly, but there is only one "reasonable" way for defining the morphisms. That is, homophisms of abelian groups $M\to N$ which are compatible with the actions of $G$ on $M$ and $N$.
Later he defined $H^1(G,M)$ where $M$ is a $G$-modules, and claimed that there is an exact sequence $$0\to M'^G \to M^G\to M''^G\to H^1(G,M')\to H^1(G,M)\to H^1(G,M'')$$ So equivalently, we can define $H^1(G,M)$ as the first derived functor of $$\begin{aligned} \square^G: G-Mod &\to \mathbf{Ab}\\ M &\to M^G \end{aligned}$$ where $G-Mod$ is what I denote for the category of $G$-modules and $\mathbf{Ab}$ is the category of abelian groups. The question is, what are the injective objects in $G-Mod$ then? Also, I 'd like to know if there are some standard reference discussing these topics.
Any help would be appreciated.