In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus $T=\mathbf{R}/\mathbf{Z}$ by subsets of $\mathbf{Z}$. Here we consider new types of subgroups: let $K\subseteq T$ be a Kronecker set (a compact set on which every continuous function $f:K\to T$ can be uniformly approximated by characters of $T$), and $G$ the group generated by $K$. We prove (Theorem 1) that $G$ can be characterized by a subset of ${\mathbf{Z}}^2$ (instead of a subset of $\mathbf{Z}$). If $K$ is finite, Theorem 1 implies our earlier result...