This is a follow up question to a previous question:
If $X$ is a transitive $G$-set, then the rank of $X$ is the number of $G_x$-orbits of $X$ where $G_x$ is the stabilizer of some $x\in X$.
If $X$ is a transitive $G$-set and $x \in X$, then rank $X$ is the number of ($G_x$-$G_x$)-double cosets in $G$.
Is there a way to understand what these double cosets that count the rank mean? The subgroup $G_x$ acts on $X$ and so the cosets of $G_x$ have meaning through that. Is there some kind of analogous context to understand double cosets?