Homotopy equivalences of p-completed classifying spaces of finite groups

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The classification of p-local finite groups over the extraspecial group of order p3 and exponent p (2004, background): “Unfortunately, a priori p-fusion is not enough to determine all the homotopy theoretic information related to BG∧p as the results in [1] show.”
Extensions of p-local nite groups (2007, background): “Thus (S, FS(G), L c (G)) is a p-local finite group, with classifying space |L c (G)| ∧ ≃ BG ∧ (see [ BLO1 , Proposition 1.1]).”
Subgroup Families Controlling p‐Local Finite Groups (2004, result): “It was shown in [4] that the homotopy theory of the nerve jLH S ðGÞj (for the right choice of H) is closely related to the p-local homotopy theory of BG.”
Extensions of linking systems with p-group kernel (2007, background): “1. In several papers, such as [ 4 ,14], the fusion and linking systems of G are shown to play a central role in describing homotopy theoretic properties of the p-completed classifying space BG ∧.”
Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups (2007, unknown): “We first recall some notation from [6] and [7].”