Let $S$ be a compact surface, and let $G$ be a nontrivial finite group acting freely on $S$ (note that this is a strong condition, since automorphisms usually have fixed points). Then $H^1(S)$ also has an action of $G$. I know how to prove, using Riemann-Hurwitz and Artin's induction theorem, that $$ H^1(S,\mathbb{Q}) \cong \mathbb{Q}^2 \oplus \mathbb{Q}[G]^{2k} $$ for some integer $k\ge 0$, where $\mathbb{Q}$ denotes the trivial $\mathbb{Q}[G]$-module.

What is the $\mathbb{Z}[G]$-module $H^1(S,\mathbb{Z})$ ?

I could not figure this out, although it feels like it should be well-known. In particular, when do we have $H^1(S,\mathbb{Z}) \cong \mathbb{Z}^2 \oplus \mathbb{Z}[G]^{2k}$ ?