In this paper we show that for every congruent monotileable amenable group and for every metrizable Choquet simplex , there exists a minimal -subshift, which is free on a full measure set, whose set of invariant probability measures is affine homeomorphic to . 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 [3] for residually finite groups.
