  Let G be a finite group and H a normal subgroup such that G/H is cyclic.
Given a conjugacy class g^G of G we define its centralizing subgroup to be
HC_G(g). Let K be such that H\le K\le 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.
