We determine the properties of the conjugate
\check{\tau} = \phi^{-1} \Delta \phi
of a diagonal function \Delta by the mark function \phi
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 Polya-Redfield enumeration.
In particular, we obtain a Polya-like substitution formula
for the K[J]-inventory of colourings of a set
whose symmetry group is the wreath product G[F].
A \beta-ring is supplied with operations \beta_H
where H runs over the conjugacy classes of subgroups
of the symmetric groups S_n.
In an earlier paper we introduced a second set of operations
\lambda_H and we show here that the two sets are related
by the isomorphism
\beta_H(-X) \cong (-1)^n \lambda_H(X).
We then consider the operations \beta_H and \lambda_H
as combinatorial species, in the sense of Joyal,
and express their molecular decomposition as a finite sum of products of
the exponential species with molecular species of degree at most n.
We give combinatorial interpretations for \beta_{S_n}-structures
and \lambda_{S_n}-structures and derive various species isomorphisms.
