Suppose we are given a reductive group $G$, its closed subgroup $H$ (not necessarily reductive), an affine $G$-variety $X$ and its closed subvariety $Y$ such that

(1) The $G$ action on $X$ is free and each orbit is closed.

(2) For $g\in G$, $gY=Y$ if and only if $g\in H$ if and only if $gY\cap Y\neq \emptyset$.

Is it true that $X//G$ is a geometric quotient of $Y$ with respect to the $H$ action?

Suppose we are given a reductive group $G$, its closed subgroup $H$ (not necessarily reductive), an affine $G$-variety $X$ and its closed subvariety $Y$ such that

(1) The $G$ action on $X$ is free and each orbit is closed.

(2) For $g\in G$, $gY=Y$ if and only if $g\in H$ if and only if $gY\cap Y\neq \emptyset$.

(3) The intersection of each $G$-orbit and $Y$ is nonempty.

Is it true that $X//G$ is a geometric quotient of $Y$ with respect to the $H$ action?

Suppose we are given a reductive group $G$, its closed subgroup $H$(not necessarily reductive), an affine $G$-variety $X$ and its closed subvariety $Y$ such that

(1) The $G$ action on $X$ is free and each orbit is closed.

(2) For $g\in G$, $gY=Y$ if and only if $g\in H$ if and only if $gY\cap Y\neq \emptyset$.

Is it true that $X//G$ is a geometric quotient of $Y$ with respect to the $H$ action.

Suppose we are given a reductive group $G$, its closed subgroup $H$  (not necessarily reductive), an affine $G$-variety $X$ and its closed subvariety $Y$ such that

(1) The $G$ action on $X$ is free and each orbit is closed.

(2) For $g\in G$, $gY=Y$ if and only if $g\in H$ if and only if $gY\cap Y\neq \emptyset$.

Is it true that $X//G$ is a geometric quotient of $Y$ with respect to the $H$ action?