Let $p:E\to X$ be a vector bundle on a smooth projective curve $X$. Let $\Gamma$ be a finite subgroup of $Aut(X)$ which has a lifted action on $E$.
How do we get an induced action of $\Gamma$ on $H^i(X,E)$?
I can see how the action could be defined for $i=0$ as these are just sections. What can we do for $i>0?$
Here's a description using Čech cohomology. An element $\alpha \in H^i(X, E)$ is represented by a choice of open cover $\{U_i\}$, together with a tuple of local sections of $E$ over each of the $(i+1)$-st order intersections of the $U_i$'s, of the form $U = U_{k_0} \cap \cdots \cap U_{k_i}$. These sections have to satisfy the cocycle conditions, and are considered up to coboundaries and up to refinement of open covers.
Now $g \in G$ acts on $\alpha$ by replacing the open cover by $\{gU_i\}$ and acting on the tuple of sections. It replaces each local section $s : U \to E|_U$ by the local section $gU \to E|_{gU}$ defined by $x \mapsto gs(g^{-1}x)$. Notice that this definition relies on the fact that $g : E \to E$ commutes with the projection $E \to X$ and $g : X \to X$. This data is the new element $g\alpha \in H^i(X, E)$.
A more abstract approach would be to say that this is ultimately about functoriality of cohomology, ignoring the details of how you wish to define it. The isomorphism $g : X \to X$ induces $H^i(X, E) \xrightarrow{\sim} H^i(X, g^*E)$. The additional data of the lift of the action to $E$ amounts to a bundle isomorphism $g^*E \xrightarrow{\sim} E$ for each $g$ (respecting multiplication), which induces $H^i(X, g^*E) \xrightarrow{\sim} H^i(X, E)$. Composing these gives $H^i(X, E) \xrightarrow{\sim} H^i(X, E)$. Unwinding these two steps gives the $x \mapsto gs(g^{-1}x)$ formula above.