  We consider families of exponential sums indexed by a subgroup of invertible
classes modulo some prime power $q$. For fixed $d$, we restrict to moduli $q$
so that there is a unique subgroup of invertible classes modulo $q$ of order
$d$. We study distribution properties of these families of sums as $q$ grows
and we establish equidistribution results in some regions of the complex plane
which are described as the image of a multi-dimensional torus via an explicit
Laurent polynomial. In some cases, the region of equidistribution can be
interpreted as the one delimited by a hypocycloid, or as a Minkowski sum of
such regions.
