Near the end of a group cohomology video lecture an example mentioned included the isomorphism of cohomology groups: Given a finite group $G$, $H^n(G; \mathbb{R}/\mathbb{Z}) \cong H^{n+1} (G;\mathbb{Z})$ for $n > 0$ as result of the Bockstein homomorphism. I do not understand why it was an isomorphism. Question: Given a torus $\mathbb{T}^m = \mathbb{R}^m/\mathbb{Z}^m$ and $G$ a finite group, does this generalize to the isomorphism $H^n(G; \mathbb{T}^m) \cong H^{n+1} (G;\mathbb{Z}^m)$ for $n > 0$ and $m > 0$?
The short exact sequence $0 \rightarrow \mathbb{Z} \hookrightarrow \mathbb{R} \twoheadrightarrow \mathbb{R}/\mathbb{Z} \rightarrow 0$ of the coefficients was mentioned in the lecture which I think is relevant. Following Example 6.11 of Rotman's "An Introduction to Homological Algebra, Second Edition", taking cohomology instead homology, and given a short exact sequence $0 \rightarrow G' \rightarrow G \rightarrow G'' \rightarrow 0$, I think there is the long exact sequence of cohomology groups $...\rightarrow H^q(X, G') \rightarrow H^q(X, G) \rightarrow H^q(X, G'') \xrightarrow{\rho_n} H^{q+1}(X, G') \rightarrow ...$ with the connecting homomorphism $\rho_n :H^q(X, G'') \rightarrow H^{q+1}(X, G')$ known as the Bockstein homomorphism. A similar question with a different short exact sequence cited Rotmam's book in the comments.
Assuming the long exact sequence is correct, there would need be the short exact sequence of $0 \rightarrow \mathbb{Z}^m \xrightarrow{\alpha} \mathbb{R}^m \xrightarrow{\beta} \mathbb{T}^m \rightarrow 0$ to the torus. I found that the torus admits a universal cover $\beta: \mathbb{R}^m \rightarrow \mathbb{T}^m$ from this answer and the map $\beta$ is surjective since it is a covering map. In the same answer the fundmental group was given as $\pi_1(\mathbb{T}^m) = \mathbb{Z}^m$. Maybe the fundamental theorem of covering space gives the injective map $\alpha$? Alternatively, if $\mathbb{Z}^m$ is a normal subgroup $\mathbb{R}^m$, this would define group extension and result in the desired short exact sequence.