Let $G$ be a finite group and $N\lhd G$. For every $x\in G-N$ we have $C_G(x)N/N\leq C_{G/N}(Nx)$ where $C_G(x)$ is the centralizer of $x$ in $G$. But I don't know what is the qutient.
In fact I want to know the $[C_{G/N}(Nx):C_G(x)N/N]$. It is more important when $N=Z(G)$.