Projective special linear group of degree two has a class-inverting automorphism

Statement

Let $k$ be a field and $G:=PSL(2,k)$ be the Projective special linear group (?) of degree two over $k$. Then, $G$ is a Group having a class-inverting automorphism (?). In other words, there is an automorphism $\sigma$ of $G$ that is a Class-inverting automorphism (?): it sends every element into the conjugacy class of its inverse.

Facts used

1. Special linear group of degree two has a class-inverting automorphism
2. Class-inverting automorphism induces class-inverting automorphism on any quotient

Proof

The proof follows directly from facts (1) and (2).