Permutation homomorphisms of block ciphers and ${\otimes _{\mathbf{W}}}$-Markovian property

We consider $\otimes$-Markov block ciphers on the alphabet $X$ with independent round keys and an Abelian group $(X, \otimes)$ of key addition. Lai X., Massey J. L., Murphy S. in 1991 had proved that the sequence of round differences of the $\otimes$-Markov block cipher forms a Markov chain. In 2017 we have given conditions under which the sequence of lumped round differences of the $\otimes$-Markov block cipher is again a Markov chain. Ciphers with such property were called ${\otimes _{\mathbf{W}}}$-Markovian block ciphers. The definition of ${\otimes _{\mathbf{W}}}$-Markovian block ciphers naturally leads to a notion of ${\otimes _{\mathbf{W}}}$-Markovian transformations. In this paper, we continue to investigate properties of ${\otimes _{\mathbf{W}}}$-Markovian ciphers. We ascertain connections between the existence of homomorphisms of block ciphers and the ${\otimes _{\mathbf{W}}}$-Markovian property.