Finite groups in which some particular invariant subgroups are TI-subgroups or subnormal subgroups

Let A and G be finite groups such that A acts coprimely on G by automorphisms. We prove that if every self-centralizing non-nilpotent A-invariant subgroup of G is a TI-subgroup or a subnormal subgroup, then every non-nilpotent A-invariant subgroup of G is subnormal and G is p-nilpotent or p-closed for any prime divisor p of | G | {|G|} . If every self-centralizing non-metacyclic A-invariant subgroup of G is a TI-subgroup or a subnormal subgroup, then every non-metacyclic A-invariant subgroup of G is subnormal and G is solvable.