On the distribution of conjugacy classes between the cosets of a finite group in a cyclic extension

Let G be a finite group and H a normal subgroup such that G/H is cyclic. Given a conjugacy class gG of G, we define its centralizing subgroup to be HCG(g). Let K be such that H ≤ K ≤ G. We show that the G-conjugacy classes contained in K whose centralizing subgroup is K are equally distributed between the cosets of H in K. The proof of this result is entirely elementary. As an application, we find expressions for the number of conjugacy classes of K under its own action, in terms of quantities relating only to the action of G.