Splitting criterion for conjugacy class in a normal subgroup

Statement
Suppose $$H$$ is a normal subgroup of a group $$G$$ and $$g \in H$$. Then, the conjugacy class of $$g$$ with respect to $$G$$ is a subset of $$H$$ that is the union of one or more conjugacy classes with respect to $$H$$. In other words, the $$G$$-conjugacy class of $$g$$ is a union of $$H$$-conjugacy classes. We can obtain a bijection:

$$H$$-conjugacy classes in the $$G$$-conjugacy class of $$g$$ $$\leftrightarrow$$ the coset space $$G/(HC_G(g))$$

where $$C_G(g)$$ denotes the centralizer of $$g$$ in $$G$$.

In particular, the $$G$$-conjugacy class of $$g$$ is a single $$H$$-conjugacy class if and only if $$HC_G(g) = G$$.

Applications

 * Splitting criterion for conjugacy classes in the alternating group
 * Splitting criterion for conjugacy classes in the special linear group