Monomial map

Definition with symbols
A function $$f:G \to G$$ is termed a monomial map if there exists a word $$w(x,y_1,y_2,\ldots,y_n)$$ and fixed elements $$a_1,a_2,\ldots,a_n \in G$$ such that for any $$g \in G$$:

$$f(g) = w(g,a_1,a_2,\ldots,a_n)$$

A monomial map which defines an endomorphism is termed a monomial endomorphism. A monomial map which defines an automorphism is termed a monomial automorphism.

Inner automorphisms
An inner automorphism is a map of the form $$g \mapsto aga^{-1}$$ where $$a \in G$$ is a fixed element. It thus fits into the definition of a monomial map. In fact, inner automorphisms are monomial automorphisms.

Universal power maps
A universal power map is a map of the form $$x \mapsto x^n$$ where $$n$$ is a fixed integer. clearly a universal power map is a monomial map, in fact, it is a monomial map with no parameters.

In particular universal power automorphisms are monomial automorphisms and universal power endomorphisms are monomial endomorphisms.