Kaleidoscopical configurations in groups

A subset $A$ of a group $G$ is called a kaleidoscopical configuration if there exists a surjective coloring $chicolon Xrightarrow kappa$ such that the restriction $chi|gA$ is a bijection for each $gin G$. We give two topological constructions of kaleidoscopical configurations and show that each infinite subset of an Abelian group contains an infinite kaleidoscopical configuration.