Let $z\in Z(G)$ then one can say that $(zx)^g=zx^g$. But it means that multiplication by $z$ create a bijection from conjugacy classes of $x$ to conjugacy classes of $xz$.
Let $\omega=\{C_1,C_2...,C_k\}$ be conjugacy classes of $G$ except the elements of $Z(G)$ then $Z(G)$ acts on $\omega$ by left multiplication.
I wonder that is there any good result based on this action ? Can we compute kernel of action or orbits of action expicitly ?