Subracks and second homology of the conjugacy classes of finite projective special linear groups of degree two

We describe the subracks of the conjugacy classes of $\mathrm{P}\mathrm{S}\mathrm{L}(2,q)$ based on Dickson’s theorem on subgroups of $\mathrm{P}\mathrm{S}\mathrm{L}(2,q)$. All minimal non-abelian subracks of $\mathrm{P}\mathrm{S}\mathrm{L}(2,q)$ are determined. Further, we provide a general result on the relationship of associated groups of conjugacy classes in perfect groups to the Schur multiplier of the group. This allows us to conclude explicit descriptions of the associated groups and the second homology of the conjugacy classes of $\mathrm{P}\mathrm{S}\mathrm{L}(2,q)$ for $q>3$.