  We prove an analog of Deligne's 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
$(k[\dd]/(\dd^2),1\ot 1 + \dd\ot \dd)$. 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
$H^i_{\rm{Sw}}(\mathcal{O}(A),K)$, $i\ge 1$, of the function algebra
$\mathcal{O}(A)$ of $A$.
