Cohomology classes of full order and restriction to subgroups

Let $G$ be a finite group, let $X$ be a representation of $G$ over $\mathbb{Z}$, and let $u \in H^2(G,X)$ be a class whose additive order equals $\#G$. Assume, for every subgroup $S \le G$ (equipped with a finite type structure), that $H^2(S, X|_S)$ is finite and $\#H^2(S, X|_S) \le \#S$, where $X|_S$ denotes the restriction of $X$ along the inclusion $S \hookrightarrow G$. Assume further given, for each subgroup $S \le G$, a $\mathbb{Z}$-linear map $\mathrm{cor}_S \colon H^2(S, X|_S) \to H^2(G,X)$ such that for all $x \in H^2(G,X)$ one has $\mathrm{cor}_S(\mathrm{res}_S x) = [G:S] \cdot x$, where $\mathrm{res}_S$ is the map induced on degree-$2$ cohomology by the inclusion $S \hookrightarrow G$ together with the identity of $X|_S$. The conclusion is twofold: first, $\#H^2(S, X|_S) = \#S$ for every subgroup $S$; second, for every subgroup $S$ the $\mathbb{Z}$-submodule of $H^2(S, X|_S)$ spanned by the single element $\mathrm{res}_S u$ is the whole module.

This is the standard passage from a degree-two class of full order, an upper bound on the orders of the $H^2$ of all subgroups, and a corestriction satisfying $\mathrm{cor}_S \circ \mathrm{res}_S = [G:S]$, to the statement that $H^2(S, X|_S)$ is cyclic of order $\#S$ generated by the restriction of that class — the cohomological shape of a fundamental class. It is used in the construction of fundamental classes for the idele class group, in particular in M4aHerbrand.exists_fundamentalClass_ideleClassGroup and its $p$-group variant.