A group $G$ is isomorphic to the fundamental groups of a compact solvmanifold if and only if it fits into the short exact sequence $1\to N\to G\to\mathbb Z^n\to 1$ where $N$ is a finitely generated torsion-free nilpotent group.
This is stated on p.253 and explained is chapter III of Auslander's <a href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-79/issue-2/An-exposition-of-the-structure-of-solvmanifolds-Part-I/bams/1183534430.full">An exposition of the structure of solvmanifolds. Part I: Algebraic theory</a>. In particular, <b> every torus bundle over a torus is a solvmanifold</b>. 

You may also be interested in Theorem 3 of Wilking's paper <a href="https://www.uni-muenster.de/imperia/md/content/theoretische_mathematik/diffgeo/mr1764235.pdf">Rigidity of group actions on solvable Lie groups</a> which gives an analogous result for infrasolvmanifolds.