Loop space homology associated with the mod 2 Dickson invariants

The spaces and have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces, one has the classifying spaces of the finite groups , for an odd prime power q, and the exotic family of classifying spaces of 2-local finite groups . In this article we compute the loop space homology of and for all odd primes q, as Hopf algebras over the Steenrod algebra, and the associated Bockstein spectral sequences.