  In this paper we show that for every congruent monotileable amenable group
$G$ and for every metrizable Choquet simplex $K$, there exists a minimal
$G$-subshift, which is free on a full measure set, whose set of invariant
probability measures is affine homeomorphic to $K$. If the group is virtually
abelian, the subshift is free. Congruent monotileable amenable groups are a
generalization of amenable residually finite groups. In particular, we show
that this class contains all the infinite countable virtually nilpotent groups.
This article is a generalization to congruent monotileable amenable groups of
one of the principal results shown in \cite{CP} for residually finite groups.
