The local lifting problem for D4

For a prime p, a cyclic-by-p group G and a G-extension L|K of complete discrete valuation fields of characteristic p with algebraically closed residue field, the local lifting problem asks whether the extension L|K lifts to characteristic zero. In this paper, we characterize D4-extensions of fields of characteristic two, determine the ramification breaks of (suitable) D4- extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every D4-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that D4 is a local Oort group for the prime 2.