Perfect diffusion of partitions of finite Abelian groups

We consider an Abelian group $(X, + )$ and study properties of permutations on $X$ acting on partitions ${{\mathbf{W}}^{(t)}}$ of ${\bar X^t}$, which consist of pairwise different elements from ${X^t}$, $t = 2,3,\ldots$ These partitions are generalizations of classical differential partitions for $t = 2$. High order differential, truncated differential, impossible differential, polytopic and multiple differential techniques use such partitions. Let $d_{{\mathbf{W}}^{(t)}}(s)$ be the minimum Hamming distance between a permutation $s$ and the set of all permutations on $X$ preserving ${{\mathbf{W}}^{(t)}}$. We describe properties of permutations $s$ with the maximal value $d_{{\mathbf{W}}^{(t)}}(s)$, which perfectly diffuse ${{\mathbf{W}}^{(t)}}$. We find a criterion of perfect diffusion of ${{\mathbf{W}}^{(t)}}$ for any $t\in \mathbb{N}$. For the additive group of a vector space over $\mathbb{F}_{2^m}$, we show the connections between permutations perfectly diffusing ${{\mathbf{W}}^{(t)}}$, APN-permutations, AB-permutations and differentially $2r$-uniform permutations, $r \ge 1$. For additive groups of vector spaces and residue rings, we also compare diffusion property of well-known $S$-boxes for ${{\mathbf{W}}^{(3)}}$.