  We consider three families of lattices on the oscillator group $G$, which is
an almost nilpotent not completely solvable Lie group, giving rise to coverings
$G \to M_{k, 0} \to M_{k, \pi} \to M_{k, \pi/2}$ for $k\in \Z$. We show that
the corresponding families of four dimensional solvmanifolds are not pairwise
diffeomorphic and we compute their cohomology and minimal models. In
particular, each manifold $M_{k, 0}$ is diffeomorphic to a Kodaira--Thurston
manifold, i.e. a compact quotient $S^1 \times \Heis_3 (\R) /\Gamma_k$ where
$\Gamma_k$ is a lattice of the real three-dimensional Heisenberg group $\Heis_3
(\R)$. We summarize some geometric aspects of those compact spaces. In
particular, we note that any $M_{k, 0}$ provides an example of a solvmanifold
whose cohomology does not depend on the Lie algebra only and which admits many
symplectic structures that are invariant by the group $\R \times\Heis_3 (\R)$
but not under the oscillator group $G$.
