We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary p-groups $$(\mathbb {Z}_p)^d$$
(
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 $$d=1, 2$$
d
=
1
,
2
, and prove some partial results for $$d=3$$
d
=
3
.