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 $(Y\times G)/H$.


Edit: forgot the assumption $H$ smooth. 



Subschemes in group action
I am writing up my comment as an answer. Let $p$ be a prime integer. Let $k$ be field of characteristic $p$. Denote by $\mathbb{A}^2\_k$ the $k$-scheme $\text{Spec}\ k[s,t]$ with its usual structure of group $k$-scheme, $$m:\mathbb{A}^2\_k \times\_{\text{Spec}\ k}\mathbb{A}^2\_k \to \mathbb{A}^2\_k, \ \ m^\*(x) = x\otimes 1 + 1\otimes x, \ \ m^\*(y) = y\otimes 1 + 1\otimes y,$$ and where the identity of this group action is the origin, i.e., the point corresponding to the maximal ideal $\mathfrak{m}=\langle x,y\rangle$. 


Denote by $\mathbb{P}^1\_k$ the $k$-scheme $\text{Proj}\ k[u,v]$ where $u$ and $v$ each have degree $1$. The group $k$-scheme $\mathbb{A}^2\_k$ induces a group $\mathbb{P}^1$-scheme by pullback, $$\text{pr}\_1:\mathbb{P}^1\_k \times\_{\text{Spec}\ k} \mathbb{A}^2\_k \to \mathbb{P}^1\_k, $$ $$(\text{pr}\_1,m\circ \text{pr}\_{2,3}):\mathbb{P}^1\_k\times\_{\text{Spec}\ k}\mathbb{A}^2\_k \times\_{\text{Spec}\ k}\mathbb{A}^2\_k \to \mathbb{P}^1\_k \times\_{\text{Spec}\ k} \mathbb{A}^2\_k.$$ Denote by $\Gamma$ the closed subscheme of $\mathbb{P}^1\_k\times\_{\text{Spec}\ k}\mathbb{A}^2\_k$ whose ideal sheaf is generated by $\text{pr}\_2^\*(\mathfrak{m}^p)$ and the equation $uy-vx$. 


The closed subscheme $\Gamma$ is a subgroup $\mathbb{P}^1$-scheme of $\mathbb{P}^1\_k\times\_{\text{Spec}\ k}\mathbb{A}^2\_k,$ that is finite and flat over $\mathbb{P}^1\_k$, and there exists a unique finite, flat morphism of group $\mathbb{P}^1$-schemes, $$q:\mathbb{P}^1\_k\times\_{\text{Spec}\ k}\mathbb{A}^2\_k \to X,$$ such that the following induced morphism is an isomorphism, $$\Gamma\times\_{\mathbb{P}^1\_k} \mathbb{A}^2\_k \to \mathbb{A}^2\_k\times\_X \mathbb{A}^2\_k.$$ This claim is easiest to prove Zariski locally on $\mathbb{P}^1\_k$. On the dense open subset $D\_+(u)\subset \mathbb{P}^1\_k$, there is an isomorphism of group $D\_+(u)$-schemes, $$\phi\_u:D\_+(u)\times\_{\text{Spec}\ k} \mathbb{A}^2\_k \to D\_+(u)\times\_{\text{Spec}\ k} \mathbb{A}^2\_k, \ \ \phi\_u^\*x = x, \ \phi\_u^\*y = y+(v/u)x.$$ The inverse image of $\Gamma$ is $D\_+(u)\times (\alpha\_p\times \{0\}),$ i.e., the copy of $\alpha\_p$ in the first factor of $\mathbb{A}^2\_k$. A quotient by this inverse image group $D\_+(u)$-scheme is the finite flat morphism of group $D\_+(u)$-schemes, $$q\_u:D\_+(u)\times \mathbb{A}^2\_k \to D\_+(u)\times \mathbb{A}^2\_k, \ \ q\_u^\*(x) = x^p, \ \ q\_u^\*(y) = y.$$ There is a similar description on $D\_+(v)$, and the uniqueness of quotients allows to glue these quotients on $D\_+(u)\cap D\_+(v)$. 


The quotient group morphism $q$ induces an action of the group $\mathbb{P}^1$-scheme $\mathbb{P}^1\_k\times\_{\text{Spec}\ k} \mathbb{A}^2\_k$ on $X$. This is equivalent to an action of $\mathbb{A}^2\_k$ on the $k$-scheme $X$ such that the projection morphism, $$\pi:X\to \mathbb{P}^1\_k,$$ is $\mathbb{A}^2\_k$-invariant. Define $Y$ to be the image of the section of $\pi$ coming from the group identity. The inverse image of $Y$ under $q$ equals $\Gamma$. Thus, the stabilizer subgroup $k$-scheme $H$ of $Y$ equals the subgroup $k$-scheme that maps $\Gamma$ to itself. 


In particular, for the fiber $\Gamma\_0$ over $\text{Zero}(v)\subset \mathbb{P}^1\_k$, this is a subgroup $k$-scheme of $\mathbb{A}^2\_k$ whose regular action on $\mathbb{A}^2\_k$ maps $\alpha\_p\times\{0\}$ to itself, i.e., a subgroup $k$-scheme of $\alpha\_p\times\{0\}$. On the other hand, for the fiber $\Gamma\_\infty$ over $\text{Zero}(u)\subset \mathbb{P}^1\_k$, this is a subgroup $k$-scheme of $\{0\}\times \alpha\_p.$ The intersection of these two subgroup $k$-schemes of $\mathbb{A}^2\_k$ is the trivial subgroup scheme $\text{Spec}\ k$. Thus, $H$ is the trivial subgroup $k$-scheme.


The induced morphism $(Y\times G)/\Delta(H) \to X$ is simply the morphism $q$. This morphism is a universal homeomorphism. However, the morphism $q$ is not an isomorphism: it is finite and flat of degree $p>1$.


