Let \(\mathbb{Z}G\) be the integral group ring of the finite group \(G\). The ‘prime graph question’ (PQ) asks whether the prime graph \(\Gamma(G)\), with vertices prime divisors of the group, \(p\) and \(q\) joined by an edge if there is an element of the group of order \(pq\), coincides with \(\Gamma(V(\mathbb{Z}G))\), a weak version of the first Zassenhaus conjecture on the conjugacy of torsion units of \(V(\mathbb{Z}G)\) with trivial units in \(\mathbb{Q}G\). (PQ) holds true for all soluble \(G\) due to the first author [Contemp. Math. 420, 215–228 (2006; Zbl 1126.20001)]. By the Hertweck-Luthar-Passi method, V. Bovdi and the second author verified (PQ) for some sporadic simple groups in a series of papers, see for instance [J. Algebra Appl. 11, No. 1, Article ID 1250016, 10 p. (2012; Zbl 1247.16032)]. There is an ongoing research by A. Bächle and L. Margolis [Proc. Edinb. Math. Soc., II. Ser. 60, No. 4, 813–830 (2017; Zbl 1380.16036)] on (PQ), they implemented the above method as the GAP package HeLP, Version 3.0 (2016).
An almost simple group of type \(S\) is a subgroup of \(\mathrm{Aut}(S)\) containing \(\mathrm{Inn}(S)\cong S\), where \(S\) is a finite non-abelian simple group. The main results of the present paper include the following. If for each almost simple group \(X\) occuring as an image of \(G\) (PQ) holds true, then (PQ) also holds true for \(G\). If the order of each almost simple group \(X\) occuring as an image of \(G\) is divisible by exactly three different primes, then (PQ) holds true for \(G\). If all composition factors of \(G\) are of prime order or isomorphic to \(\mathrm{PSL}(2,p^f)\) with \(f\in\{1,2\}\) and \(p\geq5\) then (PQ) holds true for \(G\).