Normal subgroup of ambivalent group implies every element is automorphic to its inverse

Statement
Suppose $$G$$ is an fact about::ambivalent group (i.e., every element of $$G$$ is conjugate to its inverse) and $$H$$ is a fact about::normal subgroup of $$G$$. Then, $$H$$ is a fact about::group in which every element is automorphic to its inverse: for every element $$h \in H$$, there is an automorphism of $$H$$ sending $$h$$ to its inverse.

Related facts

 * Normal subgroup of rational group implies any two elements generating the same cyclic subgroup are automorphic
 * Every element is automorphic to its inverse is characteristic subgroup-closed
 * Alternating group implies every element is automorphic to its inverse
 * General linear group implies every element is automorphic to its inverse
 * Special linear group implies every element is automorphic to its inverse
 * Projective general linear group implies every element is automorphic to its inverse
 * Projective special linear group implies every element is automorphic to its inverse