Subgroups of direct products of groups invariant under the action of permutationson factors

We study subgroups of the direct product of two groups invariant under the action of permutations on factors. An invariance criterion for the subdirect product of two groups under the action of permutations on factors is put forward. Under certain additional constraints on permutations, we describe the subgroups of the direct product of a finite number of groups that are invariant under the action of permutations on factors. We describe the subgroups of the additive group of vector space over a finite field of characteristic 2 which are invariant under the coordinatewise action of inversion permutation of nonzero elements of the field.