  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 $G$.
