Let $G$ and $G'$ be two groups and $H$ a subgroup of $G$. Take a map $f:G\rightarrow G'$ such that $f(xy)=f(x)f(y)$ for all $x\in G$ and $y\in H$. By definition, the map $f$ is not necessarily a homomorphism. So, is there a name for these kinds of maps?.
Thank you very much.
Consider the (right) action of $H \le G$ on the set $G$ by multiplication on the right, so that $G$ is an $H$-set. Consider the action of $H' = f(H) \le G'$ on the set $G'$ by multiplication on the right, so that $G'$ is an $H'$-set.
Then your situation can be described as a homomorphism of group actions, see for instance the first paragraph of this answer.