Abelian subgroup is contained in centralizer of derived subgroup in generalized dihedral group

Statement
Suppose $$H$$ is an abelian group and $$G$$ is the corresponding fact about::generalized dihedral group containing $$H$$ as an abelian normal subgroup of index two. Then, $$H$$ is contained in the fact about::centralizer of commutator subgroup $$C_G([G,G])$$. In other words, the commutator subgroup $$[G,G]$$ centralizes $$H$$.

Related facts about generalized dihedral groups

 * Abelian subgroup is isomorph-containing in generalized dihedral group unless it is an elementary abelian 2-group
 * Abelian subgroup equals centralizer of commutator subgroup in generalized dihedral group unless it is a 2-group of exponent at most four

Other facts about being contained in the centralizer of commutator subgroup

 * Commutator subgroup centralizes cyclic normal subgroup, or equivalently, any cyclic normal subgroup of a group is contained in the centralizer of commutator subgroup.
 * Commutator subgroup centralizes aut-abelian normal subgroup, or equivalently, any aut-abelian normal subgroup of a group is contained in the centralizer of commutator subgroup.
 * Abelian-quotient abelian normal subgroup is contained in centralizer of commutator subgroup

Other related facts

 * Odd-order cyclic group is characteristic in holomorph