Genuinely ramified maps and stable vector bundles

Let $f : X \rightarrow Y$ be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field, such that the corresponding homomorphism between étale fundamental groups $f_* : π_1^{\rm et}(X)\rightarrowπ_1^{\rm et}(Y)$ is surjective. Fix a polarization on $Y$ and equip $X$ with the pullback, by $f$, of this polarization on $Y$. Given a stable vector bundle $E$ on $X$, we prove that there is a vector bundle $W$ on $Y$ with $f^*W$ isomorphic to $E$ if and only if the direct image $f_*E$ contains a stable vector bundle $F$ such that $\frac{{\rm degree}(F)}{{\rm rank}(F)}= \frac{1}{{\rm degree}(f)}\cdot \frac{{\rm degree}(E)}{{\rm rank}(E)}$ We also prove that $f^*V$ is stable for every stable vector bundle $V$ on $Y$.

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