Semi-local Shapiro–Mackey injectivity in degree two for coinduced modules

Let $k$ be a commutative ring, $V$ a $k$-module, $G$ a group, and let $\Gamma$ denote the group of $\mathbb{Q}$-algebra automorphisms of $\mathrm{AlgebraicClosure}\ \mathbb{Q}$, with $r : G \to \Gamma$ a group homomorphism. Let $U \trianglelefteq \Gamma$ be a normal subgroup which is open in the sense that there is an intermediate field $F_0$ of $\mathbb{Q}$ in $\mathrm{AlgebraicClosure}\ \mathbb{Q}$, finite-dimensional over $\mathbb{Q}$, whose fixing subgroup is contained in $U$. Let $\gamma : \Gamma/(U \sqcup r(G)) \to \Gamma$ be a set-theoretic section, so $\gamma(t)$ maps to $t$ for every $t$. Let $c : G \times G \to \mathrm{Res}_r\,\mathrm{CoInd}_{U \hookrightarrow \Gamma}(V)$ be a $2$-cochain with values in the $U$-trivial module $V$ coinduced along $U \hookrightarrow \Gamma$ and then restricted along $r$, and assume $c$ lies in groupCohomology.levelCocycles₂ for $r$ and this representation. Assume further that for every $t \in \Gamma/(U \sqcup r(G))$ the map $(d,d') \mapsto c(d,d')(\gamma(t))$ on $r^{-1}(U) \times r^{-1}(U)$, obtained by evaluating the coinduced functions $\Gamma \to V$ at $\gamma(t)$, lies in groupCohomology.levelCoboundaries₂ for $r$ composed with the inclusion of $r^{-1}(U)$ and the trivial $r^{-1}(U)$-module $V$. Then $c$ itself lies in groupCohomology.levelCoboundaries₂ for $r$ and $\mathrm{Res}_r\,\mathrm{CoInd}_{U \hookrightarrow \Gamma}(V)$.

This is the injectivity half, at the level of the project's level cocycles and coboundaries, of the semi-local Shapiro–Mackey description of degree-two cohomology of $\mathrm{Res}_r\,\mathrm{CoInd}_U^{\Gamma} V$ as a product of copies of $H^2(r^{-1}(U), V)$ indexed by the double cosets $U \backslash \Gamma / r(G)$: a level $2$-cocycle all of whose components are level coboundaries is itself a level coboundary. It is used in the construction of a global class with prescribed local restrictions, through groupCohomology.exists_forall_locRes_continuousH2S_coind_trivial_eq_add_smul.