General semilinear group of degree two

Definition
Let $$K$$ be a field. The general semilinear group of degree two over $$K$$, denoted $$\Gamma L(2,K)$$, is defined as the general semilinear group of degree two over $$K$$. Explicitly, it is the external semidirect product:

$$\Gamma L (2,K) = GL(2,K) \rtimes \operatorname{Aut}(K)$$

where $$GL(2,K)$$ denotes the general linear group of degree two and $$\operatorname{Aut}(K)$$ is the group of field automorphisms of $$K$$ acting entry-wise on the matrices.

If $$k$$ is the prime subfield of $$K$$, and $$K$$ is a Galois extension of $$k$$ (note that this case always occurs for $$K$$ a finite field), then $$\operatorname{Aut}(K) = \operatorname{Gal}(K/k)$$ (the Galois group) and we get:

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