The Cohomology of Exotic 2-local Finite Groups

There exist spaces BSol(q) which are the classifying spaces of a family of 2-local finite groups based on certain fusion system over the Sylow 2-subgroups of Spin₇(q). In this paper we calculate the cohomology of BSol(q) as an algebra over the Steenrod algebra . We also provide the calculation of the cohomology algebra over of the finite group of Lie type G₂(q).