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We prove that ifL=F42⁢(22⁢n+1)′L={}^{2}F_{4}(2^{2n+1})^{\prime}and  is a nonidentity automorphism of , thenG=⟨L,x⟩G=\langle L,x\ranglehas four elements conjugate to  that generate . This result is used to study the following conjecture about the -radical of a finite group. Let  be a proper subset of the set of all primes and let  be the least prime not belonging to . Setm=rm=rifr=2r=2or 3 andm=r−1m=r-1ifr⩾5r\geqslant 5. Supposedly, an element  of a finite group  is contained in the -radicalOπ⁡(G)\operatorname{O}_{\pi}(G)if and only if every  conjugates of  generate a -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 typeB22⁢(22⁢n+1){}^{2}B_{2}(2^{2n+1}),G22⁢(32⁢n+1){}^{2}G_{2}(3^{2n+1}),F42⁢(22⁢n+1)′{}^{2}F_{4}(2^{2n+1})^{\prime},G2⁢(q)G_{2}(q), orD43⁢(q){}^{3}D_{4}(q).

On generations by conjugate elements in almost simple groups with socle 2 𝐹 4 (𝑞 2 )′