Gerbal central extensions of reductive groups by K3

We classify central extensions of a reductive group G by K3 and BK3, 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× and the group of integral Weyl-invariant cubic forms on the cocharacter lattice respectively.