Transpose-inverse map induces inner automorphism on projective general linear group of degree two

Statement

Suppose $R$ is any commutative unital ring. Let $G = GL(2,R)$ be the General linear group (?) of degree two over $R$, $Z$ be the center of $G$ (which is also the group of scalar matrices, because Center of general linear group is group of scalar matrices over center), and let $PGL(2,R) = G/Z$ be the Projective general linear group (?) of degree two over $R$.

Then, the automorphism of $PGL(2,R)$ induced by the Transpose-inverse map (?) automorphism of $G$ is an inner automorphism of $PGL(2,R)$.

Related facts

• Transpose-inverse map is inner automorphism on special linear group of degree two
• Transpose-inverse map is composite of inner automorphism and division by determinant on general linear group of degree two