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.