Suppose $f: X \to Y$ is a finite, galois map (with galois group $\Gamma$) of curves over a finite field $\mathbb F_q$. If I pick a constant sheaf $\Lambda (\cong \mathbb Z/\ell^n$, say), then $f_*(\Lambda) \cong \Lambda^\Gamma$ (by which I simply mean $\deg(f) = |\Gamma|$ many copies of $\Lambda$) since $X\times_Y X \cong \bigsqcup_{\Gamma}X$. Moreover, the Leray spectral sequence converges immediately since all the higher pushforwards are zero (since $f$ is finite) and so $$H^p(X,\Lambda) \cong H^p(Y,f_*\Lambda) \cong H^p(Y,\Lambda)^\Gamma.$$
However, I don't think this will be true. I don't think this is true even for $p=0$. Where am I going wrong?
The problem is with the statement that $f_*(\Lambda) = \Lambda^\Gamma$. While $f_*(\Lambda)(Y) = \Lambda^\Gamma$ as claimed, it is not true that the pushforward of a constant sheaf under a finite map has to be constant.
It is easy to compute the stalks of the pushforward of a finite map and we have: $$(f_*\Lambda)_y = \bigoplus_{x \to y}\Lambda_x$$ for $x,y$ geometric points on $X,Y$ and we see that over a ramified point, we have fewer points.