G $G$ -typical Witt vectors with coefficients and the norm

For a profinite group $G$ we describe an abelian group $W_G(R;M)$ of $G$-typical Witt vectors with coefficients in an $R$-module $M$ (where $R$ is a commutative ring). This simultaneously generalises the ring $W_G\left(R\right)$ of Dress and Siebeneicher and the Witt vectors with coefficients $W(R;M)$ 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 $N_{\left\{e\right\}}^G\left(X\right)$ of a connective spectrum $X$, for any finite group $G$. Our construction is reasonably analogous to the constructions of previous variants of Witt vectors, and as such is amenable to fairly explicit concrete computations.