Coinflation functor on homology

Definition
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 $$G \to G/N$$.

Related notions

 * Restriction functor on cohomology
 * Corestriction functor on homology
 * Coinflation functor on homology