We prove the Martino–Priddy conjecture for an odd prime $p$: the $p$-completions of the classifying spaces of two groups $G$ and $G^\prime$ are homotopy equivalent if and only if there is an isomorphism between their Sylow $p$-subgroups which preserves fusion. A second theorem is a description for odd $p$ of the group of homotopy classes of self homotopy equivalences of the $p$-completion of $BG$, in terms of automorphisms of a Sylow $p$-subgroup of $G$ which preserve fusion in $G$. These are both consequences of a technical algebraic result, which says that for an odd prime $p$ and a finite group $G$, all higher derived functors of the inverse limit vanish for a certain functor $\calz_G$ on the $p$-subgroup orbit category of $G$.