Gerbal central extensions of reductive groups by $K_3$

We classify central extensions of a reductive group $G$ by ${\mathcal{K}}_3$ and $B{\mathcal{K}}_3$, the sheaf of third Quillen $K$-theory groups and its classifying stack. These turn out to be parametrized by the group of Weyl-invariant quadratic forms on the cocharacter lattice valued in $k^{\times }$ and the group of integral Weyl-invariant cubic forms on the cocharacter lattice respectively.