Commuting fraction in quotient group is at least as much as in whole group

Statement
Suppose $$G$$ is a finite group and $$H$$ is a normal subgroup. Let $$K = G/H$$ be the quotient group. Then, the following are true:


 * 1) The fact about::commuting fraction (i.e., the fraction of pairs of elements that commute) in $$K$$ is at least as much as in $$G$$.
 * 2) The fact about::number of conjugacy classes in $$K$$ is bounded from below by the quotient of the number of conjugacy classes in $$G$$ by the size of $$H$$.

Similar facts

 * Commuting fraction in subgroup is at least as much as in whole group
 * Number of conjugacy classes in a subgroup may be more than in the whole group

Opposite facts

 * Number of conjugacy classes in a quotient is less than or equal to number of conjugacy classes of group