Automorphisms of kaleidoscopical graphs

A regular connected graph $\Gamma$ of degree $s$ is called kaleidoscopical if there is a $(s+1)$-coloring of the set of its vertices such that every unit ball in $\Gamma$ has no distinct monochrome points. The kaleidoscopical graphs can be considered as a graph counterpart of the Hamming codes. We describe the groups of automorphisms of kaleidoscopical trees and Hamming graphs. We show also that every finitely generated group can be realized as the group of automorphisms of some kaleidoscopical graphs.