Automorphism group of general linear group over a field

Definition
Let $$k$$ be a field and $$n$$ be a natural number. The group we are interested in is the automorphism group of the general linear group $$GL(n,k)$$ of degree $$n$$ over $$k$$. The automorphism group is denoted $$\operatorname{Aut}(GL(n,k))$$.

Structure
The structure of the automorphism group is as follows. It has a split short exact sequence:

$$1 \to \operatorname{CAut}(GL(n,k)) \to \operatorname{Aut}(GL(n,k)) \to \operatorname{Aut}(PGL(n,k)) \to 1$$.

Here, $$\operatorname{CAut}(GL(n,k))$$ is the central automorphism group of general linear group, which is given by automorphisms of the form:

$$A \mapsto A \varphi(\det A)$$

where $$\varphi$$ is a homomorphism from the multiplicative group of $$k$$ to itself chosen such that $$x \mapsto x \varphi(x^n)$$ is an automorphism of the multiplicative group of $$k$$.

$$\operatorname{Aut}(PGL(n,k))$$ is the automorphism group of projective general linear group over a field, i.e., the automorphisms of the quotient of $$GL(n,k)$$ by its center. This in turn can be expressed as:

$$PGL(n,k) \rtimes (\operatorname{Aut}(k) \times C_2)$$

where $$\operatorname{Aut}(k)$$ is the group of field automorphisms of $$k$$ with a natural induced action on $$PGL(n,k)$$, and $$C_2$$ is the cyclic group of order two acting via the transpose-inverse map.