Sylowizers and p-nilpotency in finite groups

Let [Formula: see text] and [Formula: see text] be finite groups of relative coprime orders and [Formula: see text] act on [Formula: see text] via automorphisms. An [Formula: see text]-invariant subgroup [Formula: see text] of [Formula: see text] is called an [Formula: see text]-[Formula: see text]-Sylowizer of an [Formula: see text]-invariant [Formula: see text]-subgroup [Formula: see text] if [Formula: see text] is maximal in [Formula: see text] with respect to having [Formula: see text] as its Sylow [Formula: see text]-subgroup. In this paper, we obtain some criteria for [Formula: see text]-nilpotency of a finite group by analyzing the relation to specific subsets and characteristics of the maximal [Formula: see text]-invariant subgroups of an [Formula: see text]-invariant Sylow [Formula: see text]-subgroup.