Let $\Gamma$ be a finite graph, then $H^1(\Gamma,\mathbb{Z})\cong \mathbb{Z}^{g(\Gamma)}$ can be viewed as a $\mathrm{Aut}(\Gamma)$ module.

Conversely, given a finite group $G$, and a $G$-module $\mathbb{Z}^n$, does there always exist a finite graph $\Gamma$ such that the $G$ module $\mathbb{Z}^n$ arises as $G\overset{f}{\to}\mathrm{Aut}(\Gamma)\curvearrowright H^1(\Gamma,\mathbb{Z})$ for some $f\colon G\to \mathrm{Aut}(\Gamma)$?