 Inversion of conjugacy classes

For a group $G$, inversion sends the conjugacy class of $g$ to that of $g^{-1}$, because $(xg{x}^{-1}{)}^{-1}=x{g}^{-1}{x}^{-1}$. It therefore defines an involution on the collection of conjugacy classes, even in a noncommutative group where inversion is not a group homomorphism.