On wreath product occurring as subgroup of automata group

A finitely generated group is said to be an automata group if it admits a faithful self-similar finite-state representation on some regular m-tree. We prove that if G is a subgroup of an automata group, then for each finitely generated abelian group A, the wreath product A G is a subgroup of an automata group. We obtain, for example, that C₂ (C₂ Z), Z (C₂ Z), C₂ (Z Z), and Z (Z Z) are subgroups of automata groups. In the particular case Z (Z Z), we prove that it is a subgroup of a two-letters automata group; this solves Problem 15.19 - (b) of the Kourovka Notebook proposed by A. M. Brunner and S. Sidki in 2000 [8, 17].