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 $\pi^\*F \cong E$? 



Descent of coherent sheaves on finite coverings
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 $x$.


