# Coinflation functor on homology

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 quotient of $A$ by the subgroup of elements of the form $\varphi(g)a - a$, with $g \in N$.
Then, the coinflation homomorphism $\operatorname{coinf}: H_*(G,A) \to H_*(G/N,A_N)$ is defined as the composite:
$H_*(G,A) \to H_*(G,A/N) \stackrel{\operatorname{cor}}{\to} H_*(G/N,A_N)$
where the first map is the natural map obtained by viewing cohomology as a covariant functor in its second coordinate, applied to the surjection $A \to A_N$ of $G$-modules, and the second map is obtained by applying the corestriction functor on homology to the quotient map $G \to G/N$.