Finite symmetric tensor categories with the Chevalley property in characteristic $2$

We prove an analog of Delignes theorem for finite symmetric tensor categories $mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $2$. Namely, we prove that every such category $mathcal{C}$ admits a symmetric fiber functor to the symmetric tensor category $mathcal{D}$ of representations of the triangular Hopf algebra $(k[dd]/(dd^2),1ot 1 + ddot dd)$. Equivalently, we prove that there exists a unique finite group scheme $G$ in $mathcal{D}$ such that $mathcal{C}$ is symmetric tensor equivalent to $Rep_{mathcal{D}}(G)$. Finally, we compute the group $H^2_{rm inv}(A,K)$ 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 $H^i_{rm{Sw}}(mathcal{O}(A),K)$, $ige 1$, of the function algebra $mathcal{O}(A)$ of $A$.