The étale fundamental group and F-divided sheaves in characteristic $p>0$

We investigate how the étale fundamental group controls local systems in characteristic p, namely F-divided sheaves. In analogy with Grothendieck-Malcev's results for discrete groups, we show that if a morphism f Y → X of smooth projective varieties over k=k̄ induces a surjection on the étale fundamental groups, then the pullback functor Fdiv(X)→ Fdiv(Y) is fully faithful. If f is surjective and the induced map is an isomorphism, then the functor is an equivalence. These results extend the theorem of Esnault-Mehta on the triviality of F-divided sheaves over simply connected varieties.