$k$-transitivity of a class of block transformations

Let $\Omega$ be an arbitrary finite set, $\mathcal Q(\Omega)$ be the collection of all binary quasigroups defined on the set $\Omega$, and $\Sigma^F\colon\Omega^n\to\Omega^n$ be the mapping that are implemented by a network $\Sigma$ of width $n$ with one binary operation $F\in\mathcal Q(\Omega)$. In this paper, we declare a continuation of research related to $k$-transitivity of the class $\{\Sigma^F\colon F\in\mathcal Q(\Omega)\}$ in case $k\geqslant2$. Namely, we define conditions for the $k$-transitivity of the class $\{\Sigma^F\colon F\in\mathcal Q(\Omega)\}$, propose one effective method for verification of network's $k$-transitivity for all sufficiently large finite sets $\Omega$, and give parameters of the result of the algorithm for constructing network $\Sigma$ such that the class $\{\Sigma^F\colon F\in\mathcal Q(\Omega)\}$ is $k$-transitive.