On the structure of a poly-$\mathbb{Z}$ group

In this paper we study a certain class of polycyclic groups. We outline a method for constructing a poly-$\mathbb{Z}$ group $G_n$ by describing a process for selecting maps that are used to extend $G_i$ to $G_{i+1}$ for $1 \leq i \leq n-1$ and describe the multiplicative structure and automorphism groups of some poly-$\mathbb{Z}$ groups up to $G_3$.