Suppose we have a (semi)group $G$ acting (abstractly) on a (semi)group $H$ in such a way that a mixed associativity law is satisfied: $$ g\cdot(h_1\cdot h_2)=(g\cdot h_1)\cdot h_2. $$
Equivalently, the orbit map $g\mapsto g\cdot e_H$ induces a homomorphism $G\to H$ and the action of $G$ on $H$ is the composition of this homomorphism and the left action of $H$ on itself.
Is there a name for this kind of structure? Or a reference?