Operations on Milnor-Witt K-theory

For all positive integers $n$ and all homotopy modules $M_{\ast }$, we define certain operations ${\underset{}{\mathrm{K}}}_n^{\mathrm{MW}}\to M_{\ast }$ and show that these generate the $M_{\ast }(k)$-module of all (in general non-additive) operations ${\underset{}{\mathrm{K}}}_n^{\mathrm{MW}}\to M_{\ast }$ in a suitable sense, if $M_{\ast }$ is $\mathbb{N}$-graded and has a ring structure. This also allows us to explicitly compute the abelian group $\mathrm{Op}({\underset{}{\mathrm{K}}}_n^{\mathrm{MW}},{\underset{}{\mathrm{K}}}_m^{\mathrm{MW}})$ and all operations between related theories such as Milnor, Witt and Milnor-Witt K-theory.