Wreath products in the automorphism group of a full shift

We prove that if a subgroup $H$ of the automorphism group $\text{Aut}({\mathrm{\Sigma }}^{\mathbb{Z}})$ of a non-trivial full shift acts on points of finite support (= points bi-asymptotic to a fixed point) with a free orbit, then for every finitely-generated abelian group $A$, the abstract group $A\wr H$ also embeds in $\text{Aut}({\mathrm{\Sigma }}^{\mathbb{Z}})$. The groups admitting an action with such a free orbit include $A\wr \mathbb{Z}$ for $A$ a finite abelian group, and finitely-generated free groups. The class of such groups is also closed under commensurability and direct products. We obtain for example that $\mathbb{Z}\wr \mathbb{Z}$, ${\mathbb{Z}}_2\wr ({\mathbb{Z}}_2\wr \mathbb{Z})$ and $\mathbb{Z}\wr ({\mathbb{Z}}_2\wr \mathbb{Z})$ embed in $\text{Aut}({\mathrm{\Sigma }}^{\mathbb{Z}})$. The group $\mathbb{Z}\wr \mathbb{Z}$ is the first example of a finitely-generated torsion-free subgroup of $\text{Aut}({\mathrm{\Sigma }}^{\mathbb{Z}})$ with infinite cohomological dimension, answering an implicit question of Kim and Roush and an explicit question of the author. We also explore a simpler variant of the construction that gives embeddings of certain Neumann groups, as well as some near-misses to higher iterated wreath products.