Projective semilinear group of degree two

Definition
Suppose $$K$$ is a field. The projective semilinear group of degree two over $$K$$ is defined as the defining ingredient::projective semilinear group of degree two over $$K$$. It is denoted $$P\Gamma L(2,K)$$.

It can be described as an external semidirect product of the projective general linear group of degree two over $$K$$ by the Galois group of $$K$$ over its prime subfield $$k$$, where the latter acts on the former by applying the Galois automorphism to all the matrix entries in any representing matrix:

$$P\Gamma L(2,K) = PGL(2,K) \rtimes \operatorname{Gal}(K/k)$$

In the particular case that $$K$$ is a prime field (i.e., either a field of prime size or the field of rational numbers), $$P\Gamma L(2,K)$$ can be identified with $$PGL(2,K)$$.

For a prime power $$q$$, we denote by $$P\Gamma L(2,q)$$ the group $$P\Gamma L(2,\mathbb{F}_q)$$, where $$\mathbb{F}_q$$ is the (unique up to isomorphism) field of size $$q$$.

Over finite field
We consider the case where $$K$$ is the (unique up to isomorphism) field of size $$q$$, with $$q = p^r$$, so $$p$$ is the field characteristic and $$r$$ is the order of the Galois group $$\operatorname{Gal}(K/k)$$.