In this article we analyze the essence of the recent "physical'' generalizations of classical symmetry from the viewpoint of the split extensions of groups. The principal results of the general theory of groups of Wp- and Wq-symmetry are given. The methods of deriving generalized symmetry groups of different types are formulated. All these methods are based on right, left and crossed quasi-homomorphisms and their generalizations. The principal properties of diverse quasi-homomorphic mappings are examined. The article is illustrated with many examples of colored and ïndexed" geometrical figures that are described by the groups of P-,`P-, Wp- or Wq-symmetry.