On \(p\)-radical covers of pentavalent arc-transitive graphs

Let \(\Gamma\) $\mathrm{\Gamma }$ be a finite connected pentavalent graph admitting a nonabelian simple arc-transitive automorphism group \(T\) $T$ and soluble vertex stabilizers. Let \(p>|T|_{2}\) ${p>|T|}_2$ be an odd prime and \((p,|T|)=1\) $(p,|T|)=1$ , where \(|T|_{2}\) ${|T|}_2$ is the largest power of 2 dividing the order \(|T|\) $|T|$ of \(|T|\) $|T|$ . Then we prove that there exists a \(p\) $p$ -radical cover \(\widetilde{\Gamma}\) $\tilde{\mathrm{\Gamma }}$ of \(\Gamma\) $\mathrm{\Gamma }$ such that the full automorphism group \(\text{Aut}(\widetilde{\Gamma})\) $\text{Aut}(\tilde{\mathrm{\Gamma }})$ of \(\widetilde{\Gamma}\) $\tilde{\mathrm{\Gamma }}$ is equal to \(O_{p}(\text{Aut}(\widetilde{\Gamma})).T\) $O_p(\text{Aut}(\tilde{\mathrm{\Gamma }})).T$ and the covering transformation group is \(O_{p}(\text{Aut}(\widetilde{\Gamma}))\) $O_p(\text{Aut}(\tilde{\mathrm{\Gamma }}))$ , where \(O_{p}(\text{Aut}(\widetilde{\Gamma}))\) $O_p(\text{Aut}(\tilde{\mathrm{\Gamma }}))$ is the \(p\) $p$ -radical of \(\text{Aut}(\widetilde{\Gamma})\) $\text{Aut}(\tilde{\mathrm{\Gamma }})$ .