Let $G$ be a finite group of order $n$, and let $L := (\mathbb{Q}/\mathbb{Z})^d$ be a (not necessarily trivial) $G$-module (we assume that $d$ is finite).
By a direct limit argument, there must be a finite $G$-module $M$ (in $L$) such that $$H^2(G,M) = H^2(G,L).$$
How small can I take $M$ (in terms of $G$ or even $n$)?
In particular, since $H^2(G,L)$ is $n$-torsion, can I just take $M$ to be the $n$-torsion points of $L$?
It is not clear to me why such a finite module should exist. If $m$ is an integer divisible by the order of $G$, and $M$ is $m$-torsion points in $L$ then you have an exact sequence:
$$0\rightarrow H^1(G, L)\rightarrow H^2(G, M)\rightarrow H^2(G, L)\rightarrow 0.$$
So if $H^1(G,L)\neq 0$ then the map is never an isomorphism.
Example, if $G$ is cyclic of order $n$ and $L=\mathbb{Q}/\mathbb{Z}$ with trivial $G$-action then $H^1(G,L)= \mathbb{Z}/n \mathbb Z$ and $H^2(G,L)=0$. Which $M$ do you take in this case?
