Let $X$ be a non-singular hyperelliptic curve (over $\mathbb{C}$) and $\pi:X \to \mathbb{P}^1$ be a $2:1$ covering. Let $\sigma:X \to X$ be the hyperelliptic involution and $E$ be a locally free sheaf on $X$ such that $ \sigma^* E \cong E$. Then, does there exist a locally free sheaf $F$ on $\mathbb{P}^1$ such that $\pi^*F \cong E$?
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5$\begingroup$ No: Galois descent is not faithfully flat descent for generically Galois finite flat covers that are not etale. Choose $x \in X$ at which $\pi$ is ramified and let $z=\pi(x)$, so $\{x\}$ is $\sigma$-stable. The inverse ideal sheaf $E=O(x)$ is isomorphic to its own $\sigma$-pullback (even in a manner that restricts to a descent datum over $\mathbf{P}^1-\{z\}$, which you didn't ask to be satisfied but should have required). This $E$ is not the pullback of a vector bundle $F$ on the base, as otherwise $F$ would be a line bundle and $1=\deg(E) = \deg(\pi)\deg(F)=2\deg(F)$ is even, an absurdity. $\endgroup$– nfdc23Commented Jan 31, 2018 at 4:23
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Assume $\sigma^*E \cong E$. Then $E$ is a pullback if and only if the action of $\sigma$ on the fiber of $E$ at each ramification point of $\pi$ is trivial.
In the counterexample of nfdc23, the $\sigma$ acts by $-1$ on the fiber at $x$.
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$\begingroup$ Don't you need to assume the isomorphism $f:\sigma^*(E)\simeq E$ satisfies $f\circ \sigma^*(f)= {\rm{id}}_E$ (so it is a descent datum away from the branch locus)? $\endgroup$– nfdc23Commented Feb 2, 2018 at 15:56
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$\begingroup$ @nfdc23: Of course you are right, the bundle should be equivariant. $\endgroup$– SashaCommented Feb 2, 2018 at 16:02