On Baer–Suzuki $\pi$-theorems

Given a set $\pi$ of primes, say that the Baer–Suzuki $\pi$-theorem holds for a finite group $G$ if only an element of $\mathscr O_\pi(G)$ can, together with each conjugate element, generate a $\pi$-subgroup. We find a sufficient condition for the Baer–Suzuki $\pi$-theorem to hold for a finite group in terms of nonabelian composition factors. We show also that in case $2\not\in\pi$ the Baer–Suzuki $\pi$-theorem holds for every finite group.