Injectivity of inflation on H² when H¹ of the kernel vanishes

Let $G$ and $G'$ be finite groups, let $\pi\colon G'\to G$ be a surjective group homomorphism, let $C$ be a $\mathbb{Z}$-linear representation of $G$ and $C'$ one of $G'$, and let $j\colon \mathrm{Res}_\pi C\to C'$ be a morphism of representations of $G'$, where $\mathrm{Res}_\pi C$ is $C$ with $G'$ acting through $\pi$; thus the underlying $\mathbb{Z}$-linear map $j$ satisfies $j(\rho_C(\pi a)c)=\rho_{C'}(a)j(c)$ for all $a\in G'$, $c\in C$. Assume that $j$ is injective on underlying modules; that every $c'\in C'$ fixed by every element of $\ker\pi$ lies in the image of $j$; and that the degree-one group cohomology of the restriction of $C'$ along the inclusion $\ker\pi\hookrightarrow G'$, i.e. $H^1(\ker\pi, C')$, is a zero object. The conclusion is that the underlying map of the morphism $H^2(G,C)\to H^2(G',C')$ induced by the pair $(\pi, j)$, namely groupCohomology.map π j 2, is injective.

This is the injectivity (inflation) half of the inflation–restriction exact sequence in degree two, in the form in which the coefficient module upstairs has its $\ker\pi$-invariants captured by $j$ and $H^1$ of the kernel vanishes. It is used in the computation of fundamental classes of $S$-idèle class groups, via M4aHerbrand.exists_fundamentalClass_ideleClassGroup_map_eq_finrank_smul_of_ne_two.