On groups of reversible automata

For any nontrivial finite group $\mathbb{X}$ we construct a reversible automaton whose state set and alphabet coincide with $\mathbb{X}$, with transition and output functions defined by the left regular action of $\mathbb{X}$. We then study the associated automaton group $G_{\mathbb{X}}$ and prove, in particular, that $G_{\mathbb{X}}$ contains the lamplighter group $\mathbb{A} \wr \mathbb{Z}$, where $\mathbb{A}$ denotes the center of $\mathbb{X}$.