# Inflation functor on cohomology

Suppose $G$ is a group and $N$ is a normal subgroup of $G$. Suppose $A$ is an abelian group and $\varphi:G \to \operatorname{Aut}(A)$ is a homomorphism of groups, making $A$ into a $G$-module. Denote by $A^N$ the subgroup of $A$ fixed pointwise by all elements of $N$.
Then, the inflation homomorphism $\operatorname{inf}: H^*(G/N;A^N) \to H^*(G;A)$ is defined as the composite:
$H^*(G/N;A^N) \stackrel{\operatorname{res}}{\to} H^*(G;A^N) \to H^*(G;A)$
where the first map is the restriction homomorphism $H^*(G/N;A^N) \to H^*(G;A^N)$ corresponding to the quotient map $G \to G/N$ and the second map is the natural map obtained by viewing cohomology as a covariant functor in its second coordinate, applied to the injection $A^N \to A$ of $G$-modules.