  Let $p:\Sigma'\to\Sigma$ be a finite Galois cover, possibly branched, with
Galois group $G$. We are interested in the structure of the cohomology of
$\Sigma'$ as a module over $G$. We treat the cases of branched and unbranched
covers separately. In the case of branched covers, we give a complete
classification of possible module structures of $H_1(\Sigma',\bC)$. In the
unbranched case, we algebro-geometrically realize the representation of $G$ on
holomorphic 1-forms on $\Sigma'$ as functions on unions of $G$-torsors. We also
explicitly realize these representations for certain branched covers of
hyperelliptic curves. We give some applications to the study of pseudo-Anosov
homeomorphisms of surfaces and representation theory of the mapping class
group.
