We develop a fragment of the theory of Duflo-Serganova functor over a field of odd characteristic. We elaborate a method of computing the symmetry supergroup \(\widetilde{\mathbb {G}_x}\) of this functor, recently introduced by A.Sherman, for a wide class of supergroups \(\mathbb {G}\) , and apply it to the case when \(\mathbb {G}\) is \(\textrm{GL}(m|n)\) or \(\textrm{Q}(n)\) , and a square zero odd element \(x\in \textrm{Lie}(\mathbb {G})\) has minimal or maximal rank. For any quasi-reductive supergroup \(\mathbb {G}\) , which has a pair of specific parabolic supersubgroups, we prove a criterion of injectivity of a \(\mathbb {G}\) -supermodule, involving vanishing of Duflo-Serganova functor on it.
