Class-inverting automorphism induces class-inverting automorphism on any quotient

This article gives the statement, and possibly proof, of a group property (i.e., group having a class-inverting automorphism) satisfying a group metaproperty (i.e., quotient-closed group property)
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Statement

Statement with symbols

Suppose $G$ is a group having a class-inverting automorphism. In other words, there is a Class-inverting automorphism (?) $\sigma$ of $G$: an automorphism $\sigma$ such that for all $g\in G$, $\sigma (g)$ is conjugate to $g^{-1}$. Suppose $N$ is a normal subgroup of $G$. Then, $G/N$ is also a group having a class-inverting automorphism. In fact, $\sigma (N)=N$ and the automorphism induced by $\sigma$ on $G/N$ is a class-inverting automorphism.