6.7 (1978)

Suppose that $P$ is a finite 2-group. Does there exist a characteristic subgroup $L(P)$ of $P$ such that $L(P)$ is normal in $H$ for every finite group $H$ that satisfies the following conditions:
$\qquad$ 1) $P$ is a Sylow 2-subgroup of $H$,
$\qquad$ 2) $H$ is $S_4$-free, and
$\qquad$ 3) $C_H(O_2(H)) \leqslant O_2(H)$?