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

This article gives the statement, and possibly proof, of a group property (i.e., group in which every element is automorphic to its inverse) satisfying a group metaproperty (i.e., characteristic subgroup-closed group property)
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Statement

Suppose $G$ is a 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