Nonabelian key addition groups and $\otimes _{\mathbf{W}}$-markovian property of block ciphers

For an Abelian key addition group $\left( {X, \otimes } \right)$ and a partition ${\bf{W}} = \{ {W_0},\ldots ,{W_{r-1}}\}$ of a set $X$ we had introduced ${ \otimes _{\bf{W}}}$-markovian transformations and ${ \otimes _{\bf{W}}}$-markovian ciphers. The ${ \otimes _{\bf{W}}}$-markovian condition is required to validate different generalizations of differential technique. In this paper, we study ${ \otimes _{\bf{W}}}$-markovian ciphers and transformations on an nonabelian group $\left( {X, \otimes } \right)$. We get restrictions on the structure of groups $(X, \otimes )$, $\left\langle {{g_k}|k \in X} \right\rangle $ and blocks of a nontrivial partition ${\bf{W}}$ as a consequence of the condition of partial preservation of $\bf{W}$ by the round function ${g_k}\colon X \to X$ for all $k \in X$. For all nonabelian groups of the order ${2^m}$ with a cyclic subgroup having index $2$ we describe classes of ${ \otimes _{\bf{W}}}$-markovian permutations.