Finite symmetric tensor categories with the Chevalley property in characteristic 2

We prove an analog of Deligne's theorem for finite symmetric tensor categories C with the Chevalley property over an algebraically closed field k of characteristic 2. Namely, we prove that every such category C admits a symmetric fiber functor to the symmetric tensor category D of representations of the triangular Hopf algebra (k[d]/(d(2)), 1 circle times 1 + d circle times d). Equivalently, we prove that there exists a unique finite group scheme G in D such that C is symmetric tensor equivalent to Rep(D)(G). Finally, we compute the group H-inv(2) (A, K) of equivalence classes of twists for the group algebra K[A] of a finite abelian p-group A over an arbitrary field K of characteristic p > 0, and the Sweedler cohomology groups H-Sw(i) (O( A), K), i >= 1, of the function algebra O( A) of A.