Finite-state self-similar actions of nilpotent groups

Let G be a finitely generated torsion-free nilpotent group and $${\phi:H\rightarrow G}$$ be a surjective homomorphism from a subgroup H < G of finite index with trivial $${\phi}$$ -core. For every choice of coset representatives of H in G there is a faithful self-similar action of the group G associated with $${(G, \phi)}$$. We are interested in what cases all these actions are finite-state and in what cases there exists a finite-state self-similar action for $${(G, \phi)}$$. These two properties are characterized in terms of the Jordan normal form of the corresponding automorphism $$\widehat{\phi}$$ of the Lie algebra of the Mal’cev completion of G.