On generations by conjugate elements in almost simple groups with socle $\text{mbox}{}^2{F}_4(q^2{)}^{\mathrm{\prime }}$

We prove that if $L=\text{mbox}{}^2{F}_4(2^{2n+1}{)}^{\mathrm{\prime }}$ and $x$ is a nonidentity automorphism of $L$ then $G=⟨L,x⟩$ 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 $r⩾5$. Supposedly, an element $x$ of a finite group $G$ is contained in the $\pi$-radical ${\mathrm{O}}_{\pi }(G)$ 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 ${}^2{B}_2(2^{2n+1})$, ${}^2{G}_2(3^{2n+1})$, ${}^2{F}_4(2^{2n+1}{)}^{\mathrm{\prime }}$, $G_2(q)$, or ${}^3{D}_4(q)$.