Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of a point in $Y$ and both are themselves Lie groups?

Equivalently, what conditions need to be satisfied by $X$ and $Y$ for it to be possible to realise $X$ and $Y$ as simultaneous coset spaces $G/H_1$ and $G/H_2$ of $G$ with $H_1\leq H_2$?

This question is related: Simultaneous coset spaces.

Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of a point in $Y$ and both are themselves Lie groups?

Equivalently, what conditions need to be satisfied by $X$ and $Y$ for it to be possible to realise $X$ and $Y$ as simultaneous coset spaces $G/H_1$ and $G/H_2$ of $G$ with $H_1\leq H_2$?

This question is related: Simultaneous coset spaces.