The spaces BG_2 and BDI(4) 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 for q odd the classifying spaces of the finite groups
BG_2(q)and the exotic family of classifying spaces of 2-local finite groups
BSol(q). In this article compute the mod 2 loop space homology of the
2-completed classifying space of G_2(q) and of BSol(q) for all odd primes q, as
algebras over the Steenrod algebra, and the associated Bockstein spectral
sequences.