Abelian-quotient abelian normal subgroup is contained in centralizer of derived subgroup

Statement
Suppose $$G$$ is a group and $$H$$ is an fact about::abelian normal subgroup of $$G$$ that is also an fact about::abelian-quotient subgroup, i.e., the quotient group $$G/H$$ is an abelian group. Then, $$H$$ is contained in the fact about::centralizer of commutator subgroup $$C_G([G,G])$$.

Related facts

 * Commutator subgroup centralizes cyclic normal subgroup, hence any cyclic normal subgroup is contained in the centralizer of commutator subgroup.
 * Commutator subgroup centralizes aut-abelian normal subgroup, hence any aut-abelian normal subgroup is contained in the centralizer of commutator subgroup.
 * Abelian subgroup is contained in centralizer of commutator subgroup in generalized dihedral group
 * Abelian subgroup equals centralizer of commutator subgroup in generalized dihedral group unless it is a 2-group of exponent at most four