Number of conjugacy classes in extension group is bounded by product of number of conjugacy classes in normal subgroup and quotient group

Statement in terms of conjugacy classes
 Suppose $$G$$ is a finite group and $$H$$ is a normal subgroup of $$G$$ with quotient group $$G/H$$. Denote by $$c(G),c(H),c(G/H)$$ respectively the number of conjugacy classes in $$G,H,G/H$$ respectively. Then, we have the relation:

$$c(G) \le c(H)c(G/H)$$

Statement in terms of conjugacy classes
 Suppose $$G$$ is a finite group and $$H$$ is a normal subgroup of $$G$$ with quotient group $$G/H$$. Denote by $$CF(G),CF(H),CF(G/H)$$ respectively the commuting fractions of $$G,H,G/H$$ respectively. Then, we have the relation:

$$CF(G) \le CF(H)CF(G/H)$$