Lamplighter groups, bireversible automata and rational series over finite rings

We realize lamplighter groups A≀ℤ, with A a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize A≀ℤ as a bireversible automaton group if and only if the 2-Sylow subgroup of A has no multiplicity one summands in its expression as a direct sum of cyclic groups of order a power of 2.