Continuous Shapiro isomorphism in degree two for open S

Let $k$ be a commutative ring and $G$ a group, let $r \colon G \to \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be a group homomorphism into the group of $\mathbb{Q}$-algebra automorphisms of AlgebraicClosure ℚ, and let $S \le G$ be a subgroup subject to the openness hypothesis hS: there is an intermediate field $F_0$ of $\overline{\mathbb{Q}}/\mathbb{Q}$, finite-dimensional over $\mathbb{Q}$, whose fixing subgroup has $r$-preimage contained in $S$. Let $N$ be a $k$-linear representation of $S$ (an object of Rep k S). Then the type of $k$-linear equivalences between $\mathrm{continuousH2}\,r\,(\mathrm{Rep.coind}\ S.\mathrm{subtype}\ N)$ and $\mathrm{continuousH2}\,(r \circ S.\mathrm{subtype})\,N$ is nonempty. Here, for a level map $r$ and a representation $M$, $\mathrm{continuousH2}\ r\ M$ is the quotient of the submodule levelCocycles₂ r M of inhomogeneous $2$-cochains by the preimage in it of levelCoboundaries₂ r M; the first argument is formed for the representation of $G$ coinduced from $N$ along the inclusion $S \hookrightarrow G$, and the second for $N$ with the level map obtained by restricting $r$ to $S$. Only the existence of such an equivalence is asserted, no particular map being named in the conclusion.

This is Shapiro's lemma in degree two for the continuous (level-wise) cohomology used in the project: coinduction from an open subgroup $S$ does not change $H^2$. It is used in the inductive proof of the local Euler–Poincaré characteristic identity and in the finite-dimensionality of $H^2_{\mathrm{cts}}$ in the prime-local setting.