epimorphism of fppf sheaves is an fppf morphism

I asked this question on math.stackexchange (https://math.stackexchange.com/questions/2693471/epimorphism-of-fppf-sheaves-is-an-fppf-morphism) but didn't get an answer. Maybe someone here can help.

Suppose $0→F→G→H→0$ is an exact sequence of group schemes (over some base scheme $S$ ) by which I mean that the corresponding sequence of fppf-sheaves is exact.

I read somewhere that the surjectivity of the sequence of fppf-sheaves is equivalent to the fact that the morphism $G\to H$ is fppf (faithfully flat and locally of finite presentation).

I was able to prove one of the directions: If $G\to H$ is fppf then the sequence of fppf-sheaves is surjective.

I'm not sure how to prove the other direction. Any help/reference would be much appreciated.

• That is not correct. Consider the case that $H$ is the trivial group $S$-scheme, i.e., just $S$ as an $S$-scheme, and $F=G$ is an arbitrary group $S$-scheme that is not necessarily flat over $S$. – Jason Starr Mar 20 '18 at 10:20
• The result is correct if you assume, in addition, that $F$ is flat over $S$, since then $G$ is an $F$-torsor over $H$. – Jason Starr Mar 20 '18 at 10:25