The paper under review continues the study of the problem of classifying finite symmetric tensor categories \(\mathcal C\) over an algebraically closed field \(k\) of characteristic \(p>0\). More precisely, the authors classify symmetric integral tensor categories \(\mathcal C\) over \(k\) which have the Chevalley property (i.e., categories in which the tensor product of every two simple objects is semisimple). In the main result of the paper (Theorem 1.1) the authors show that every finite symmetric integral tensor category \(\mathcal C\) with the Chevalley property over an algebraically closed field \(k\) of characteristic \(p>2\) admits a symmetric fiber functor to the category of supervector spaces. This result implies a conjecture giving by Ostrik, and as a Corollary the authors obtain that every finite symmetric unipotent tensor category \(\mathcal C\) over an algebraically closed field \(k\) of characteristic \(p>2\) admits a symmetric fiber functor to the category of vector spaces. Moreover, the authors classify finite dimensional triangular quasi-Hopf algebras with the Chevalley property over an algebraically closed field \(k\) of characteristic \(p>2\) (Theorem 3.1) and certain finite dimensional triangular Hopf algebras with the Chevalley property (Section 4). The paper finishes with an appendix by Coulembier and Etingof which gives another proof of the above classifications results and shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic \(p\neq 2\) is always a Serre subcategory.