  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 ${}^3D_4(q)$.
