Let $G$ be a group and $H,K$ be subgroups of $G$ such that $G=<H,K>$. Suppose that $H,K$ acts on the set $S$.
Is there any condition that which guarantees that action of $H,K$ is extended to action of $G$ ?
What I mean can we define action of $G$ by the actions of $H,K$ ? Probably we need some compability conditions. What are they ?