Let $G$ and $H$ be groups, both acting on a set $X$. Suppose that there is a homomorphism $\phi:G\to H$ such that for every $g\in G$ and $x\in X$, $g\cdot x = \phi(g)\cdot x$. Is there a name for this particular situation? What about in the case that $\phi$ is a surjection/quotient map? The latter is a particular case of orbit equivalence of actions, but seems to deserve a stronger name.