Let $X$ be a scheme and $S$ be a sheaf of sets over the fppf topology of $X$. Let $G$ be a group scheme over $X$ and there is an action of $G$ on $S$. Now, I want to look at the quotient $G \setminus S$. The way I want to define quotient is $ (G\setminus S)(U) = G(U) \setminus S(U) $? Is it necessary that $(G\setminus S)$ is a sheaf? To define $(G\setminus S)$ is necessary to do specification?

Now, suppose $(G\setminus S)$ is representable. Does that imply $S$ is representable?