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

Statement

Suppose $G$ is a group and $H$ is an Abelian normal subgroup (?) of $G$ that is also an Abelian-quotient subgroup (?), i.e., the quotient group $G/H$ is an abelian group. Then, $H$ is contained in the 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