Transpose-inverse map is class-preserving automorphism for general linear group of degree two iff field has size two

Statement
The transpose-inverse map for the general linear group of degree two over a field $$F$$ is a class-preserving automorphism if and only if the field $$F$$ is field:F2.

Facts used

 * 1) uses::Transpose-inverse map is class-inverting automorphism for general linear group
 * 2) uses::General linear group of degree two is ambivalent iff field has size two

Proof
The proof follows directly from the stated facts.