Let $A$ and $B$ be groups and $N\unlhd A$ is a normal subgroup of $A$. Suppose that $B$ acts on $A$; that is, there exists a group homomorphism (not necessarily monomorphism) $$f:B\to\mathrm{Aut}(A)\text{,}$$ where $\mathrm{Aut}(A)$ is the automorphism group of $A$.
My question: Under what condition does $f$ descends to an action on the factor group $A/N$? That is, under what condition does $f$ induces a group homomorphism $$\overline{f}:B\to\mathrm{Aut}(A/N)\text{?}$$