There are two groups, $G_1$ and $G_2$. They are both acting on a set $S$. 

$S$ may have some structure. The groups may too. The actions respect it.

$G_1$ is mysterious. Perhaps all we know about it is the way it acts on $S$. We'd like to know more.

$G_2$ is well-known. Its structure and its $S$-action are totally transparent.

We might be able to learn about $G_1$ from watching how its action interacts with the action of $G_2$. For instance, how their orbits overlap and intersect might tell about subgroups.

Is this sort of thing systematically studied under some name? Beyond the case when the actions commute.