Conjugacy-closed implies focal subgroup equals derived subgroup

Statement
Suppose $$H$$ is a conjugacy-closed subgroup of a group $$G$$. In other words, any two elements of $$H$$ that are conjugate in $$G$$ are in fact conjugate in $$H$$. Then, the focal subgroup of $$H$$ equals its fact about::commutator subgroup.

Related facts

 * Focal subgroup theorem
 * Conjugacy-closed and Sylow implies retract