In earlier work the authors determined the graded Witt kernel GW(q)(E/F) = ker(GW(q)F -> GW(q)E) when E/F is a biquadratic extension in characteristic 2 by calculating the cohomological kernel H-2*(E/F) = ker(H-2*F -> H-2*E). In this paper this result is extended to the cases where [E : F] = 4 and E is either cyclic or has dihedral Galois closure. In addition, the use of Izhboldin's Q-groups is generalized to obtain six-term exact sequences that describe the behavior of these graded rings whenever four-term exact sequences and homotopies describe the arithmetic of the extension. These tools are valid in characteristic p, although the applications here are in characteristic 2. (C) 2016 Published by Elsevier Inc. Keywords. Author Keywords:Quadratic forms; Differential forms; Witt groups; Izhboldin groups