Cycle indices and subgroup lattices

We determine the properties of the conjugate of a diagonal function \Delta by the mark function in the incidence algebra of the poset of conjugacy classes of subgroups of a finite group G. Particular choices for \Delta provide applications to the Burnside ring of G, to the theory of \beta-rings and to Redfield-Polya enumeration. In particular, we obtain a Polya-like substitution formula for the K[J]-inventory of colourings of a set whose symmetry group is a wreath product G[F].