Simple groups and Sylow subgroups

Firstly a problem to characterize Sylow $2$-subgroups (of small order $\leqslant 2^{10}$) of finite simple groups is proposed. Next some (possibly necessery) reduction steps are discussde. The latter half of these notes is devoted to finite groups having extra-spacial $p$-groups of order $p^3$ as Sylow subgroups.