Class-inverting automorphism implies every element is automorphic to its inverse

Statement
Suppose $$G$$ is a group having a fact about::class-inverting automorphism, i.e., there exists an automorphism $$\sigma$$ of $$G$$ such that for every $$g \in G$$, there exists $$x \in G$$ such that $$\sigma(g) = xg^{-1}x^{-1}$$. Then, $$G$$ is a fact about::group in which every element is automorphic to its inverse: for every $$g \in G$$, there exists an automorphism $$\alpha$$ of $$G$$ such that $$\alpha(g) = g^{-1}$$.

Related facts

 * General linear group implies every element is automorphic to its inverse
 * Special linear group implies every element is automorphic to its inverse
 * Alternating group implies every element is automorphic to its inverse
 * Every element is automorphic to its inverse is characteristic subgroup-closed
 * Normal subgroup of ambivalent group implies every element is automorphic to its inverse