Suppose we have a curve $X$ and a group $G$ acting on $X$. Then one has an induced action of $G$ on the sheaf cohomology of $\mathcal O_X$. I wondered what one can say about the group action on the Čech cohomology of $\mathcal O_X$.
I assume that if $G$ does not change the cover then it is fine, and each element of $G$ can just act on each part of the product forming the Čech cohomology separately.
If, however, the group does not fix the cover then is there anything one can do (other than potentially refine the cover)? I can't seem to find any mention of group actions on Čech cohomology at all, so I don't know if I am asking the wrong thing.
To clarify what I mean by "changing the cover", I mean that for all $g \in G$ and $U \in \mathcal U$, where $\mathcal U$ is the cover, we have $gU \subset U$.
[Also, if I have use the wrong "Č" for "Čech cohomology", could someone please let me know what is the correct way to type it]