The lamplighter group of rank two generated by a bireversible automaton

We construct a 4-state 2-letter bireversible automaton generating the lamplighter group $({\mathbb{Z}}_2^2)\wr \mathbb{Z}$ of rank two. The action of the generators on the boundary of the tree can be induced by the affine transformations on the ring ${\mathbb{Z}}_2[[t]]$ of formal power series over ${\mathbb{Z}}_2$.