A β-ring is supplied with operations βH where Hruns over the conjugacy classes of subgroups of the symmetric groups Sn. In an earlier paper we introduced a second set of operations λH and we show here that the two sets are related by the isomorphism

formula here

We then consider the operations βH and λH 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 βSn-structures and λSn-structures and derive various species isomorphisms.