Finite-state self-similar actions of nilpotent groups

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