Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. Let $c(H)$ denote the number of conjugacy classes of a finite group $H$.
(a) Is it true that $c(G)\leq c(N)c(G/N)$?
(b) When does equality hold?
(c) Is the ratio $c(N)c(G/N)/c(G)$ always an integer?
This question can also be phrased in terms of a short exact sequence of finite groups $$1\longrightarrow N\longrightarrow G\longrightarrow K\longrightarrow1.$$ If this short exact sequence is isomorphic to $1\to N\to N\times K\to K\to1$ then it can be shown that $c(G)=c(N)c(K)$. However, even if the short exact sequence splits (meaning that the short exact sequence is isomorphic to $1\to N\to N\rtimes K\to K\to1$), it might not be the case that $c(G)=c(N)c(K)$. For instance, $N$ and $K$ might both be abelian while $G$ is nonabelian.