Let $G$ and $H$ be groups acting on sets $M$ and $N$. Suppose that there is a group homomorphism $\varphi:G\to H$ and a map $F:M\to N$ such that $$F(g\cdot p)=\varphi(g)\cdot F(p)$$ for all $p\in M$ and $g\in G$. How do we call such a map? Is there a standard terminology?
When $G=H$ and $\varphi$ is the identity we say that $F$ intertwines the actions, or that $F$ is equivariant. Is it standard to use either of these words in the more general case? Like "$\varphi$-equivariant", or something similar?
It's simply an equivariant map, but the target is a bit different. Given an $H$-set $N$ and a morphism $\varphi : G \to H$, you get a $G$-set $\varphi^* N$; it has the same underlying set as $N$, and the action is prescribed by $g \cdot n := \varphi(g) \cdot n$. Then $F$ is by definition a $G$-equivariant map $M \to \varphi^* N$. If the morphism $\varphi$ is implied from the context, then you can omit it from the notation if you want.