Characteristic subgroups and factorization

Question

Let $G$ be a finite group, $C$ and $N$ be characteristic subgroups of $G$.

Is it true that $CN/N$ is a characteristic subgroup of $G/N$?

I try to answer this question in the case, where $N$ is a center of $G$. But it seems to me that it is also interesting for arbitrary characteristic subgroup $N$.

Answer 1

Don't you have a counterexample with $G = D_{8}$, the dihedral group of order 8. When $C = C_{4}$, the cyclic group of order 4, and $N=Z(G)$.