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Classification and decomposition of the Witt–Burnside ring and Burnside ring of a profinite group

Let G be a profinite group and q be an integer. In this paper, we investigate the structure of the Witt–Burnside ring functor BqG and the generalized Burnside ring functor BGq, which are introduced in Oh [‘q‐deformation of Witt–Burnside rings’, Math. Z. 257 (2007) 151–191]. More precisely, we prove a classification theorem as q ranges over the set of integers and a decomposition theorem over the category of binomial rings when G is given by a direct sum of finitely many finite groups whose orders are mutually relatively prime. Also connections between the ring of truncated Witt vectors and Witt–Burnside rings will be dealt with.