Group homomorphism between quotient groups. Let $G$ and $H$ be two groups and $M, N$ be normal subgroups of $G$ and $H$ respectively. Consider a relation $\phi: G \to H$ be a homomorphism.
Now, my question is

what is the necessary condition on $\phi$ such that it induces a homomorphism between $\frac{G}{M}$ and $\frac{H}{N}.$


The map $\pi\circ \phi\colon G\to H/N$ factors through $G/M$ if and only if $M\subseteq \mathrm{ker}(\pi\circ\phi)$. The kernel of $\pi\circ\phi$ is $\phi^{-1}(\ker(\pi)) = \phi^{-1}(N)$.
So a necessary and sufficient condition is $M\subseteq \phi^{-1}(N)$.
