Action on orbits defined by another action

Question

Let $(G,\Omega)$ be a transitive action of $G$ on $\Omega$. Let now $H\le G$ be a subgroup of $G$ and define $\Omega/H=\{\omega\cdot H|\omega\in\Omega\}$ as the set of the orbits generated by the action of $H$ on $\Omega$. I've already showed that $$(\omega\cdot H)\cdot g=(\omega\cdot g)\cdot H,\text{ }\forall\omega\cdot H\in \Omega/H\text{ and }g\in G. $$is a group action of $G$ on $\Omega/H$.
I've problems in defining the stabilizer of a generic $\omega\cdot H\in\Omega/H$ in the action described.
Could you give me some advice? Thank you.

Answer 1

If $G$ acts on $\Omega $ transitively then, choosing some $\omega _0$ in $\Omega $, and letting $K$ be the stabilizer of $\omega _0$, the map $$ g\in G\mapsto \omega _0g\in \Omega $$ factors through the quotient $ K\backslash G = \{Kg: g\in G\}, $ providing a equivariant bijection $$ K\backslash G \to \Omega , $$ so we may assume WLOG that $\Omega =K\backslash G$.

Now, the orbit space of the right action of the given subgroup $H$ on $K\backslash G$ consists of the set of all "double cosets" $$ \Omega /H = K\backslash G/H = \{KgH: g\in G\}. $$

Thus, unless $H$ or $K$ are normal, as proposed by @Berci, there is no sensible action of $G$ on that double coset space. The fundamental parameter "$g$" in $KgH$, protected on the left and right by $K$ and $H$, is too inaccessible to be acted upon in any meaningful way.