Invariance of continuous H⁰, H¹, H² under isomorphic data

Let $k$ be a commutative ring and let $G$, $H$ be groups, all three in a single universe. Suppose given level maps, i.e. group homomorphisms $r_G \colon G \to (\overline{\mathbb{Q}} \simeq_{\mathrm{alg}[\mathbb{Q}]} \overline{\mathbb{Q}})$ and $r_H \colon H \to (\overline{\mathbb{Q}} \simeq_{\mathrm{alg}[\mathbb{Q}]} \overline{\mathbb{Q}})$ into the group of $\mathbb{Q}$-algebra automorphisms of AlgebraicClosure ℚ, a group isomorphism $e \colon G \simeq H$ compatible with them in the sense that $r_H(e(g)) = r_G(g)$ for all $g \in G$, representations $N_G \in \mathrm{Rep}\,k\,G$ and $N_H \in \mathrm{Rep}\,k\,H$, and a $k$-linear equivalence $\varphi \colon N_G \simeq N_H$ of the underlying modules satisfying $\varphi(\rho_{N_G}(g)x) = \rho_{N_H}(e(g))(\varphi(x))$ for all $g \in G$, $x \in N_G$. The conclusion is the conjunction of three nonemptiness assertions: the $k$-modules of invariants $\rho_{N_G}$-invariants and $\rho_{N_H}$-invariants admit a $k$-linear equivalence; the submodule continuousH1 rG NG of $H^1(G,N_G)$, defined as the image of levelCocycles₁ rG NG under the projection H1π, admits a $k$-linear equivalence with continuousH1 rH NH; and the quotient continuousH2 rG NG of levelCocycles₂ rG NG by the preimage in it of levelCoboundaries₂ rG NG admits a $k$-linear equivalence with continuousH2 rH NH. Only existence of such equivalences is asserted, not a canonical choice.

This is transport of structure for continuous (level-constant) cohomology in degrees $0$, $1$ and $2$ along an isomorphism of triples (group, level map, module). It is used to identify the continuous cohomology of one and the same group presented in two different ways, and is invoked in the proofs that the comparison maps $\theta_0$, $\theta_1$, $\theta_2$ and the dual-twist map are bijective under an openness hypothesis.