5.12 (1976)

Let $G$ be a finite group with trivial soluble radical in which there are Sylow 2-subgroups having non-trivial intersection. Suppose that, for any two Sylow 2-subgroups $P$ and $Q$ of $G$ with $P \cap Q \neq 1$, the index $|P : P \cap Q|$ does not exceed $2^n$. Is it then true that $|P| \leqslant 2^{2n}$?