Abelian subgroup is isomorph-containing in generalized dihedral group unless it is an elementary abelian 2-group

Statement
Suppose $$G$$ is the fact about::generalized dihedral group corresponding to an abelian group $$H$$ that is not an elementary abelian 2-group, i.e., the exponent of $$H$$ is not $$1$$ or $$2$$. Then, $$H$$ is an fact about::isomorph-containing subgroup of $$G$$.

Related facts

 * 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
 * Commutator subgroup centralizes cyclic normal subgroup
 * Commutator subgroup centralizes aut-abelian normal subgroup
 * Odd-order cyclic group is characteristic in holomorph