  We formulate and prove a new variant of the Segal Conjecture describing the
group of homotopy classes of stable maps from the p-completed classifying space
of a finite group G to the classifying space of a compact Lie group K as the
p-adic completion of the Grothendieck group of finite principal (G,K)-bundles
whose isotropy groups are p-groups. Collecting the result for different primes
p, we get a new and simple description of the group of homotopy classes of
stable maps between (uncompleted) classifying spaces of groups. This
description allows us to determine the kernel of the map from the Grothendieck
group A(G,K) of finite principal (G,K)-bundles to the group of homotopy classes
of stable maps from BG to BK.
