# Self-similar group action

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.