Inflation functor on cohomology

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 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.

Related notions

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