A coprime action version of a solubility criterion of Deskins (Q667554)

The main goal of this short well-written article is to provide a proof of the following result: Theorem: Let G and A be finite groups of co-prime orders, and assume that A acts on G by automorphisms. If G has a maximal A-invariant subgroup that is nilpotent with a Sylow 2-subgroup of class less than 3, then G is soluble. This result is in the spirit of a solubility criterion of W.E.Deskins who showed that a finite group possessing a maximal subgroup that is nilpotent and of class less than 3 is soluble [\textit{W. E. Deskins}, Ill. J. Math. 5, 306--313 (1961; Zbl 0097.25701)]. The proof uses the classification of finite simple groups.