Generalized dihedral groups are ambivalent

Statement
Let $$H$$ be any abelian group and $$G$$ be the generalized dihedral group corresponding to $$H$$. Then, $$G$$ is an ambivalent group: every element of $$G$$ is conjugate to its inverse. In particular, if $$G$$ is finite, all characters of $$G$$ are real-valued.

Related facts about generalized dihedral groups

 * Generalized dihedral groups are strongly ambivalent
 * Classification of rational generalized dihedral groups

Related facts about similar groups

 * Dihedral groups are ambivalent
 * Dicyclic groups of even degree are ambivalent
 * Symmetric groups are rational
 * Classification of ambivalent alternating groups

Proof outline
Any element in the abelian normal subgroup is conjugate to its inverse via the conjugating element. An element outside the abelian normal subgroup is an involution -- it has order two, so it is conjugate to its inverse for obvious reasons.