Outer semilinear group

Definition
Suppose $$K$$ is a field with prime subfield $$k$$and $$n$$ is a natural number. The outer semilinear group of degree $$n$$ over $$K$$, denoted $$O\Gamma L(n,K)$$, is defined as the external semidirect product:

$$GL(n,K) \rtimes (\operatorname{Gal}(K/k) \times \mathbb{Z}/2\mathbb{Z})$$

where $$\operatorname{Gal}(K/k)$$ acts coordinate-wise on the matrix entries by Galois automorphisms and the non-identity element of $$\mathbb{Z}/2\mathbb{Z}$$ acts by the transpose-inverse map. Note that the two actions commute with each other, so we can combine these to get the action of the external direct product.