Every element is automorphic to its inverse is characteristic subgroup-closed

Statement
Suppose $$G$$ is a fact about::group in which every element is automorphic to its inverse. Suppose $$H$$ is a characteristic subgroup of $$G$$. Then, $$H$$ is also a group in which every element is automorphic to its inverse.

Related facts

 * Normal subgroup of ambivalent group implies every element is automorphic to its inverse
 * Alternating group implies every element is automorphic to its inverse
 * General linear group implies every element is automorphic to its inverse
 * Projective general linear group implies every element is automorphic to its inverse
 * Special linear group implies every element is automorphic to its inverse
 * Projective special linear group implies every element is automorphic to its inverse