A natural question in the algebraic theory of quadratic forms is the determination of Witt kernels, i.e. the kernel of the restriction map when passing from the Witt ring or Witt group of a field to that of a field extension. In general, this is a difficult problem. For odd degree field extensions, the Witt kernels are zero due to a theorem of Springer. For degree 2 extensions, Witt kernels have been known for quite some time (in any characteristic). For degree 4 extensions, these kernels have been determined completely by Sivatski in characteristic not 2. We determine Witt kernels for degree 4 extensions in characteristic 2, extending the partial results that have been known so far. In characteristic 2, there is an added difficulty because of possible inseparability of the extensions leading.