On finite groups all of whose non-abelian maximal invariant subgroups of order divisible by p are TI-subgroups

Let [Formula: see text] and [Formula: see text] be finite groups such that [Formula: see text] acts coprimely on [Formula: see text] by automorphisms. For any fixed prime divisor [Formula: see text] of [Formula: see text], we provide a complete characterization of the structure of group [Formula: see text] all of whose non-abelian maximal [Formula: see text]-invariant subgroups of order divisible by [Formula: see text] are TI-subgroups.