Remarks on symmetric fusion categories of low rank in positive characteristic

We give lower bounds for the rank of a symmetric fusion category in characteristic $p\ge 5$ in terms of $p$. We prove that the second Adams operation ${\psi }_2$ is not the identity for any non-trivial symmetric fusion category, and that symmetric fusion categories satisfying ${\psi }_2^a={\psi }_2^{a-1}$ for some positive integer $a$ are super Tannakian. As an application, we classify all symmetric fusion categories of rank 3 and those of rank 4 with exactly two self dual simple objects.