I'm reading Cohomology of Number Fields by Neukirch et al. 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? If yes: is there a standard construction of such a C given G?