On cosets in the direct product of troups whose images by bijective mappings from factors to groups are cosets

This paper is devoted to the images of cosets in the direct product of groups by bijective mappings from factors to groups. We prove necessary and sufficient conditions on bijective mappings for existence a coset in the direct product of two groups whose image is a coset. Cosets in the direct product of groups, whose images by bijective mappings from factors to groups are cosets, are described with some constraints on bijective mappings. Cosets in the direct product of elementary abelian 2-groups, whose images by multiplicative inverse permutation on factors are cosets, are described. Also cosets in the direct product of elementary abelian 2-groups, whose images by $s$-box of Kuznyechik, are described. Automorphisms of the direct product groups, which commute with bijective mappings from factors to groups, are described with some constraints on bijective mappings.