  We introduce a notion of partition wreath product of a finite group by a
partition quantum group, a construction motivated on the one hand by classical
wreath products and on the other hand by the free wreath product of J. Bichon.
We identify the resulting quantum group in several cases, establish some of its
properties and show that when the finite group in question is abelian, the
partition wreath product is itself a partition quantum group. This allows us to
compute its representation theory, using earlier results of the first named
author.
