Vector bundles trivialized by proper morphisms and the fundamental group scheme, II

Let $X$ be a projective and smooth variety over an algebraically closed field $k$. Let $f:Y\to X$ be a proper and surjective morphism of $k$-varieties. Assuming that $f$ is separable, we prove that the Tannakian category associated to the vector bundles $E$ on $X$ such that $f^{\ast }E$ is trivial is equivalent to the category of representations of a finite and etale group scheme. We give a counterexample to this conclusion in the absence of separability.