On generations by conjugate elements in almost simple groups with socle \(\mbox{}^2F_4(q^2)'\)

We prove that if \(L=\mbox{}^2F_4(2^{2n+1})'\) and \(x\) is a nonidentity automorphism of \(L\) then \(G=\langle L,x\rangle\) has four elements conjugate to \(x\) that generate \(G\). This result is used to study the following conjecture about the \(\pi\)-radical of a finite group: Let \(\pi\) be a proper subset of the set of all primes and let \(r\) be the least prime not belonging to \(\pi\). Set \(m=r\) if \(r=2\) or \(3\) and set \(m=r-1\) if \(r\geqslant 5\). Supposedly, an element \(x\) of a finite group \(G\) is contained in the \(\pi\)-radical \(\operatorname{O}_\pi(G)\) if and only if every \(m\) conjugates of \(x\) generate a \(\pi\)-subgroup. Based on the results of this paper and a few previous ones, the conjecture is confirmed for all finite groups whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, or unitary simple group, or to one of the groups of type \({}^2B_2(2^{2n+1})\), \({}^2G_2(3^{2n+1})\), \({}^2F_4(2^{2n+1})'\), \(G_2(q)\), or \({}^3D_4(q)\).