This is a repost from stackexchange. I didn't get an answer, so I figured I'd ask it here.

Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ on $X$, and suppose that $Y$ is $H$ stable, that is $h.y \in Y$ for all $h\in H$ and $y\in Y$. You can form the quotient spaces $H\setminus Y$ and $G \setminus X$, and there is a natural, continuous, in general neither injective nor surjective map $\theta : H\setminus Y\rightarrow G\setminus X$. I am looking for conditions that assure this is a homeomorphism.

You can show easily that $\theta$ is onto $\mathrm{iff}~Y$ intersects all orbits, and one to one $\mathrm{iff} ~ \forall y\in Y, H.y=G.y\cap Y$. So I'll suppose these two conditions.

$\mathrm{QUESTION:}$ When is $\theta$ a homeomorphism?

All spaces $X$ I have in mind are Hausdorff, but not necessarily locally compact. Also, the groups $G$ I consider are Lie groups, but I am interested in weaker conditions too, and don't want to restrict myself to that case. I am looking for practical $sufficient$ conditions on $X,Y,H,G$ and $\rho$.

One way to make $\theta$ into a homeomorphism is to have compact (Hausdorff) $Y$ and $H$, and Hausdorff $G \setminus X$.

References would be perfect!