Let $X$ be a scheme over a separably closed field, equipped with an action of a constant group scheme $G$.
Let $H$ be a finite group whose size is invertibile on $X$, and $Y\to X$ an $H$-torsor with respect to the étale topology.



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> Is $Y\to X$ equipped with a $G$-equivariant structure?
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Étale torsors and equivariant structures
If $G$ fixes a basepoint $x\in X(k)$, then your question roughly asks whether $G$ acts trivially on (the prime-to-$p$ completion of) $\pi\_1(X, x)$. This suggests a negative answer.


For example, let $X=\mathbf{G}\_m^2$, $G=\mathbf{Z}/2$ acting on $X$ by swapping the coordinates, let $H=\mu\_\ell(k)$, and let $Y\to X$ be the pull-back of the Kummer torsor $\ell\colon\mathbf{G}\_m\to \mathbf{G}\_m$ along the first projection.


In contrast, if $G$ was a connected algebraic group, the answer should be positive. Indeed, such a $G$ can only act trivially on the finite set $H^1(X, H)$.


