  We consider a finite group $G$ with a normal subgroup $N$ so that all
elements of $G \setminus N$ have prime power order. We prove that if there is a
prime $p$ so that all the elements in $G \setminus N$ have $p$-power order,
then either $G$ is a $p$-group or $G = PN$ where $P$ is a Sylow $p$-subgroup
and $(G,P,P \cap N)$ is a Frobenius-Wielandt triple. We also prove that if all
the elements of $G \setminus N$ have prime power orders and the orders are
divisible by two primes $p$ and $q$, then $G$ is a $\{ p, q \}$-group and $G/N$
is either a Frobenius group or a $2$-Frobenius group. If all the elements of $G
\setminus N$ have prime power orders and the orders are divisible by at least
three primes, then all elements of $G$ have prime power order and $G/N$ is
nonsolvable.
