On spaces of self-homotopy equivalences of p-completed classifying spaces of finite groups and homotopy group extensions

Fix a prime p. A mod-p homotopy group extension of a group π by a group G is a fibration with base space Bπp∧ and fibre BGp∧. In this paper we study homotopy group extensions for finite groups. We observe that there is a strong analogy between homotopy group extensions and ordinary group extensions. The study involves investigating the space of self-homotopy equivalences of a p-completed classifying space. In particular, we show that under the appropriate assumption on G, the identity component of this space is homotopy equivalent to BZ(G), the classifying space of the centre of G. We proceed by studying the group of components. We show that this group maps into a group of natural equivalences of a certain functor with kernel and cokernel, which are computable in terms of the first and second derived functors of the inverse limit for a certain diagram of abelian groups.