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

Statement

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

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