I asked the following question on Stackexchange and got no reply so I am reposting it here. Let $G$ be a finite group. A $G$-module C is a **class module** if, for all subgroups $H \subset G$:

1) $H^1(H,C)=0$

2) $H^2(H,C)$ is cyclic of order $\#H$

Remark: If $G$ is cyclic then $\mathbb{Z}$ is a class module.

** Question:** Does every finite group admit a class module? Abelian group? If yes: is there a standard construction of such a C given G?