Genuinely ramified maps and pseudo-stable vector bundles

Let X and Y be irreducible normal projective varieties, of same dimension, defined over an algebraically closed field, and let f : Y -> X be a finite generically smooth morphism such that the corresponding homomorphism between the etale fundamental groups f(*) : pi(et)(1)(Y) -> pi(et)(1)(X) is surjective. Fix a polarization on X and equip Y with the pulled-back polarization. For a point y(0) is an element of Y, let pi(Y, y(0)) (resp. pi(X, f(y(0)))) be the affine group scheme given by the neutral Tannakian category defined by the strongly pseudo-stable vector bundles of degree zero on Y (resp. X). We prove that the homomorphism pi(Y, y(0)) -> pi(X, f(y(0))) induced by f is surjective. Let E be a pseudo-stable vector bundle on X and F subset of f * E a pseudo-stable subbundle with mu(F) = mu(f * E). We prove that f * E is pseudo-stable and there is a pseudo-stable subbundle W subset of E such that f * W = F as subbundles of f * E.