Ambivalence is quotient-closed

Statement
If $$G$$ is an ambivalent group, and $$H$$ is a normal subgroup of $$G$$, the quotient group $$G/H$$ is also an ambivalent group.

Related facts

 * Rationality is quotient-closed

Proof
Given: An ambivalent group $$G$$, a normal subgroup $$H$$ of $$G$$ with quotient map $$\varphi:G \to G/H$$, and an element $$u \in G/H$$.

To prove: There exists an element $$v \in G/H$$ such that $$vuv^{-1} = u^{-1}$$.

Proof: