  We discuss the relation of tiling, weak tiling and spectral sets in finite
abelian groups. In particular, in elementary $p$-groups $(\mathbb{Z}_p)^d$, we
introduce an averaging procedure that leads to a natural object of study: a
4-tuple of functions which can be regarded as a common generalization of tiles
and spectral sets. We characterize such 4-tuples for $d=1, 2$, and prove some
partial results for $d=3$.
