Special linear group of degree two has a class-inverting automorphism

Statement
Let $$k$$ be a field and $$G := SL(2,k)$$ be the fact about::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.

We can choose the following to be the class-inverting automorphism: conjugation in $$GL(2,k)$$ by the matrix $$\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}$$. Alternatively, we can choose conjugation in $$GL(2,k)$$ by the matrix $$\begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}$$.

Related facts

 * Projective special linear group of degree two has a class-inverting automorphism
 * Special linear group of degree two is ambivalent iff -1 is a square
 * Projective special linear group of degree two is ambivalent iff -1 is a square
 * Transpose-inverse map is class-inverting automorphism for general linear group
 * Transpose-inverse map induces class-inverting automorphism on projective general linear group