The paper under review contains several characterisations of finite \(\pi\)-soluble groups by means of properties of subgroups belonging to concrete families of maximal subgroups which are defined using the index and normal index, the latter concept introduced by W. E. Deskins [Proc. Symp. Pure Math. 1, 100–104 (1959; Zbl 0096.24801)]. A central concept in this paper is the one of \(\theta\)-subgroup introduced by Y. Zhao [J. Pure Appl. Algebra 124, No. 1–3, 325–328 (1998; Zbl 0893.20008)].
Typical result is the following: Theorem 3.3. Let \(G\) be a \(p\)-soluble group and \(\pi\) a set of primes. Then \(G\) is \(\pi\)-soluble if for each maximal subgroup \(M\) of \(G\) whose index is composite and normal index is not divisible by \(p\), there exists a normal \(\theta\)-subgroup \(C\) for \(M\) such that \(C/ \text{Core}_{G}(M \cap C)\) is \(\pi\)-soluble.