The one-prime power hypothesis for conjugacy classes restricted to normal subgroups and quotient groups

We say that a group G satisfies the one-prime power hypothesis for conjugacy classes if the greatest common divisor for all pairs of distinct conjugacy class sizes are prime powers. Insoluble groups which satisfy the one-prime power hypothesis have been classified. However it has remained an open question whether the one-prime power hypothesis is inherited by normal subgroups and quotients groups. In this note we construct examples to show the one-prime power hypothesis is not necessarily inherited by normal subgroups or quotient groups.