Commutator of element and automorphism

Basic definition
Let $$G$$ be a group, $$a \in G$$ be an element, and $$\sigma \in \operatorname{Aut}(G)$$ be an automorphism of $$G$$. Then, the commutator of $$a$$ and $$\sigma$$, denoted $$[a,\sigma]$$ is sometimes defined as (for those following the left-action convention):

$$a\sigma(a)^{-1}$$

Under the right-action convention, the commutator is written as $$[\sigma,a]$$ and is defined as:

$$\sigma(a)^{-1}a$$

The notion of commutator gives the usual notion of commutator of two elements $$a$$ and $$b$$, if we take $$\sigma$$ as conjugation by $$b$$ (left and right notions respectively).

Definition as an ordinary commutator of elements
Let $$G$$ be a group, $$a \in G$$ be an element and $$\sigma \in \operatorname{Aut}(G)$$. Consider the external semidirect product $$G \rtimes \langle \sigma \rangle$$ (we can take the semidirect product of $$G$$ with any subgroup of $$\operatorname{Aut}(G)$$ containing $$\sigma$$). The commutator of $$a$$ and $$\sigma$$ is simply the commutator of these as elements in the semidirect product.