Dear all,

For a finite group $G$, let $m(G)$ denote the largest number of irreducible characters of the same degree of $G$. You can say that $m(G)$ is the largest multiplicity of character degrees of $G$. I wonder if the following is true: $$m(G)\leq |N|m(G/N) \text{ for every normal subgroup } N \text{ of } G.$$ For my purpose, I only need the inequality for cyclic $N$. But I strongly believe that the inequality is good in general.

Do you see a counterexample or an idea of proof?