GG‐typical Witt vectors with coefficients and the norm

For a profinite group we describe an abelian group of ‐typical Witt vectors with coefficients in an ‐module (where is a commutative ring). This simultaneously generalises the ring of Dress and Siebeneicher and the Witt vectors with coefficients of Dotto, Krause, Nikolaus and Patchkoria, both of which extend the usual Witt vectors of a ring. We use this new variant of Witt vectors to give a purely algebraic description of the zeroth equivariant stable homotopy groups of the Hill–Hopkins–Ravenel norm of a connective spectrum , for any finite group . Our construction is reasonably analogous to the constructions of previous variants of Witt vectors, and as such is amenable to fairly explicit concrete computations.