Groups Having all Elements off a Normal Subgroup with Prime Power Order

We prove that if Gis a finite group, Nis a normal subgroup, and there is a prime pso that all the elements in G∖ Nhave p-power order, then either Gis a p-group or G= PNwhere Pis a Sylow p-subgroup and (G,P,P∩ N) is a Frobenius–Wielandt triple. We also prove that if all the elements of G∖ Nhave prime power orders and the orders are divisible by two primes pand q, then Gis a {p,q}-group and G/Nis either a Frobenius group or a 2-Frobenius group. If all the elements of G∖ Nhave prime power orders and the orders are divisible by at least three primes, then all elements of Ghave prime power order and G/Nis nonsolvable.