Let \(\Gamma \) be a connected 7-valent symmetric Cayley graph on a finite non-abelian simple group G. If \(\Gamma \) is not normal, Li et al. (J Algebraic Combin 56:1097–1118, 2022) characterized the group pairs \((\textrm{soc}(\textrm{Aut}(\Gamma )/K),GK/K)\) , where K is a maximal intransitive normal subgroup of \(\textrm{Aut}(\Gamma )\) . In this paper, we improve this result by proving that if \(\Gamma \) is not normal, then \(\textrm{Aut}(\Gamma )\) contains an arc-transitive non-abelian simple normal subgroup T such that \(G<T\) and \((T,G)=(\textrm{A}_{n},\textrm{A}_{n-1})\) with \(n=7\) , \(3\cdot 7\) , \(3^2\cdot 7\) , \(2^2\cdot 3\cdot 7\) , \(2^3\cdot 3\cdot 7\) , \(2^3\cdot 3^2\cdot 5\cdot 7\) , \(2^4\cdot 3^2\cdot 5\cdot 7\) , \(2^6\cdot 3\cdot 7\) , \(2^7\cdot 3\cdot 7\) , \(2^6\cdot 3^2\cdot 7\) , \(2^6\cdot 3^4\cdot 5^2\cdot 7\) , \(2^8\cdot 3^4\cdot 5^2\cdot 7\) , \(2^7\cdot 3^4\cdot 5^2\cdot 7\) , \(2^{10}\cdot 3^2\cdot 7\) , \(2^{24}\cdot 3^2\cdot 7\) . Furthermore, \(\textrm{soc}(\textrm{Aut}(\Gamma )/R)=(T\times R)/R\) , where R is the largest solvable normal subgroup of \(\textrm{Aut}(\Gamma )\) .
