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\begin{document}$${\varGamma }$$\end{document} be a connected cubic graph admitting a vertex-transitive almost simple group G of automorphisms. In this paper, we study the normality of the socle T of G in the full automorphism group Aut(Γ)\documentclass[12pt]{minimal}
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\begin{document}$$\text {Aut}({\varGamma })$$\end{document} of Γ\documentclass[12pt]{minimal}
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\begin{document}$${\varGamma } $$\end{document}. It is proved that if T is not normal in Aut(Γ)\documentclass[12pt]{minimal}
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\begin{document}$$\text {Aut}({\varGamma })$$\end{document}, then T=A47\documentclass[12pt]{minimal}
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\begin{document}$$T= \text {A}_{47}$$\end{document}, A23\documentclass[12pt]{minimal}
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\begin{document}$$\text {A}_{23}$$\end{document}, A2f-1\documentclass[12pt]{minimal}
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\begin{document}$$\text {A}_{2^f-1}$$\end{document} with f≥3\documentclass[12pt]{minimal}
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\begin{document}$$f\ge 3$$\end{document}, or a simple group of Lie type of even characteristic with some exceptions. In particular, if Γ\documentclass[12pt]{minimal}
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\begin{document}$${\varGamma }$$\end{document} is arc-transitive and T is not normal in Aut(Γ)\documentclass[12pt]{minimal}
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\begin{document}$$\text {Aut}({\varGamma })$$\end{document}, then Γ\documentclass[12pt]{minimal}
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\begin{document}$${\varGamma } $$\end{document} is a Cayley graph on G, and (Aut(Γ),G)=(A48,A47)\documentclass[12pt]{minimal}
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\begin{document}$$(\text {Aut}({\varGamma }),G)=(\text {A}_{48},\text {A}_{47})$$\end{document} or (S24,S23)\documentclass[12pt]{minimal}
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\begin{document}$$(\text {S}_{24},\text {S}_{23})$$\end{document}.
