Orbital derivatives over subgroups and their combinatorial and group-theoretic properties

Properties of the orbital derivatives over subgroups of the group ${{G}_{n}}$ generated by the additive groups of the residue ring ${{\mathbb{Z}}_{{{2}^{n}}}}$ and the $n$-dimensional vector space ${{V}_{n}}$ over the field $GF(2)$ are considered. Nonrefinable sequences of nested orbits for the subgroups of the group ${{G}_{n}}$ and of the Sylow subgroup ${{P}_{n}}$ of the symmetric group ${{S}_{{{2}^{n}}}}$ are described. For the orbital derivatives, three analogs of the concept of the degree of nonlinearity for functions over ${{\mathbb{Z}}_{{{2}^{n}}}}$ or ${{V}_{n}}$ are suggested.