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 fact about::projective special linear group of degree two over $$k$$. Then, $$G$$ is a fact about::group having a class-inverting automorphism. In other words, there is an automorphism $$\sigma$$ of $$G$$ that is a fact about::class-inverting automorphism: it sends every element into the conjugacy class of its inverse.

Facts used

 * 1) uses::Special linear group of degree two has a class-inverting automorphism
 * 2) uses::Class-inverting automorphism induces class-inverting automorphism on any quotient

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