Let $k$ be a field, of any characteristic. Let $G$ be a smooth group scheme over $k$ and let $X$ be a smooth scheme of finite type over $k$. Let $Y\subseteq X$ be a smooth subscheme of $X$, and let $H\subseteq G$ be the group subscheme stabilizing $Y$ (i.e. $gY(R)=Y(R)$ for every $g\in H(R)$ and every $k$-algebra $R$). Assume that the map $Y\times^HG\to X$ is geometrically injective and surjective. Does this imply that this is an isomorphism, even in characteristic $p$ (in characteristic zero this is true much more generally, thanks to ZMT)? In all the concrete examples I have, this is true, but my examples are quite mild. One can assume that $X$ is connected, but $Y$ might not be.

By $Y\times^HG$ I mean $(Y\times G)/H$.