  Given a finite group G and a field F, a G-set X gives rise to an
F[G]-permutation module F[X]. This defines a map from the Burnside ring of G to
its representation ring over F. It is an old problem in representation theory,
with wide-ranging applications in algebra, number theory, and geometry, to give
explicit generators of the kernel K_F(G) of this map, i.e. to classify pairs of
G-sets X, Y such that F[X] is isomorphic to F[Y]. When F has characteristic 0,
a complete description of K_F(G) is now known. In this paper, we give a similar
description of K_F(G) when F is a field of characteristic p>0 in all but the
most complicated case, which is when G has a subquotient that is a
non-p-hypo-elementary (p,p)-Dress group.
