On spectra of automorphic extensions of finite simple groups \(F_4(q)\) and \(^3D_4(q)\) (Q2816942)

Let \(G\) be a finite group. The spectrum \(\omega(G)\) (or denoted by \(\pi_e(G)\)) is the set of element orders of \(G\). Let \(S\) be one of the groups \(F_4(q)\), \(^3D_4(q)\). The authors describe the spectra of \(S\) and determine \(G\) such that \(S \leq G \leq\mathrm{Aut}(S)\) and \(\omega(G) = \omega(S)\). Combining this result with earlier work (see [\textit{H. P. Cao} et al., Sib. Mat. Zh. 45, No. 6, 1256--1262 (2004; Zbl 1079.20020); translation in Sib. Math. J. 45, No. 6, 1031--1035 (2004); \textit{V. D. Mazurov}, Algebra Logic 52, No. 5, 400--403 (2013; Zbl 1329.20015); translation from Algebra Logika 52, No. 5, 601--605 (2013)]), the authors determine all finite group \(G\) with \(\omega(G) = \omega(S)\).