Self-similar group action

Definition

Suppose $G$ is a group and $X$ is a finite set which we think of as an alphabet. Let $X^{*}$ be the set of all words on $X$ (viewed as a monoid), and let $\cdot$ denote a faithful group action of $G$ on $X^{*}$ as a set (in particular, the action is not an action by monoidal automorphisms). We say that $\cdot$ is a self-similar group action if for every $g\in G$ and $x\in X$, there exist $h\in G$ and $y\in X$ such that:

$g\cdot (xw)=y(h\cdot w)$

The multiplication by concatenation is happening in the monoid $X^{*}$, where it is literally symbol concatenation.

The term self-similar group is typically used for a group that admits a self-similar group action, but in its usual usage, the action is understood to be implicitly specified, even though it is not part of the abstract group structure.