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

Statement with symbols
Suppose $$G$$ is a group having a class-inverting automorphism. In other words, there is a fact about::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.