On generations by conjugate elements in almost simple groups with socle 2퐹4(푞2)′.

We prove that if L = F 4 2 ⁢ (2 2 ⁢ n + 1) ′ and 푥 is a nonidentity automorphism of 퐿, then G = ⟨ L , x ⟩ has four elements conjugate to 푥 that generate 퐺. This result is used to study the following conjecture about the 휋-radical of a finite group. Let 휋 be a proper subset of the set of all primes and let 푟 be the least prime not belonging to 휋. Set m = r if r = 2 or 3 and m = r − 1 if r ⩾ 5 . Supposedly, an element 푥 of a finite group 퐺 is contained in the 휋-radical O π ⁡ (G) if and only if every 푚 conjugates of 푥 generate a 휋-subgroup. Based on the results of this and previous papers, the conjecture is confirmed for all finite groups whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, unitary simple group, or to one of the groups of type B 2 2 ⁢ (2 2 ⁢ n + 1) , G 2 2 ⁢ (3 2 ⁢ n + 1) , F 4 2 ⁢ (2 2 ⁢ n + 1) ′ , G 2 ⁢ (q) , or D 4 3 ⁢ (q) . [ABSTRACT FROM AUTHOR]