  In this note, we prove that the F-fundamental group scheme is birational
invariant for smooth projective varieties. We prove that the F-fundamental
group scheme is naturally a quotient of the Nori fundamental group scheme. For
elliptic curves, it turns out that the F-fundamental group scheme and the Nori
fundamental group scheme coincides. We also consider an extension of the Nori
fundamental group scheme in positive characteristic using semi-essentially
finite vector bundles and prove that in this way, we do not get a non-trivial
extension of the Nori fundamental group scheme for elliptic curves, unlike in
characteristic zero.
