Let $G$ be a group, and let $M$ and $N$ be "$G$-modules", i.e. abelian groups equipped with an action of $G$. Let $f:M\to N$ be an additive map which is not quite $G$-equivariant, but rather, satisfies the following identity for all $m\in M$: $$ f(g\cdot m) = \phi(g) \cdot g \cdot f(m) , $$ where $\phi:G\to G$ is a fixed group homomorphism, independent of $m$.
First of all: does this property have a name, something like "pseudo-equivariant"?
Second: let $M/G$ and $N/G$ be the quotients under the action of $G$. Define the following map $f':M/G\to M/G$: $$ f(Gm) := Gf(m). $$ This map is additive, and well-defined, even if $f$ is not equivariant. More in general, what is the most general property that a map $f$ needs to satisfy in order to descend to the quotients?
(This question is similar in spirit to this other one.)