Suppose I have a curve $X$ (for concreteness, we can take $X$ to be a smooth, projective curve over a finite field $\mathbb F_q$, and even more concretely consider the family of curves described by the equations $y^2 = f(x^n)$), and consider $H^1(X)$ using your favorite cohomology theory (etale, singular, de Rham, crystalline...) with integral coefficients ($\mathbb Z_\ell, \mathbb Z...$). The group $G = \mathrm{Aut}(X)$ acts on $H^1(X)$, and I am interested in it's structure as a $G$-rep.
If I were instead working with rational coefficients, I could use the Lefschetz fixed point theorem applied to $g \in G$ to compute the trace of $g$ on $H^1(X) \otimes \mathbb Q$ in terms of the fixed points of $g$.This is one way to prove the Chevalley-Weil theorem. But character theory doesn't work for integral representations (or does it?)...
I would be happy with any tools or ideas that work even in special cases. I would also be happy with a qualitative statement that, for instance, describes the integral $G$-action explicitly on a sublattice of bounded index.
