In a previous paper (2) I defined the fundamental group σ(X, x0, G) of a group Gof homeomorphisms of a space X, and showed that if the transformation group admits a family of preferred paths, then σ(X, x0, G) can be represented as a group extension of π1(X, x0) by G. In this paper the homotopy groups of a transformation group are defined. The nth absolute homotopy group of a transformation group which admits a family of preferred paths is shown to be representable as a split extension of the nth absolute torus homotopy group τn(X, x0) by G.

In § 6 it is shown that the action of G on X induces a homomorphism of Ginto a quotient group of a subgroup of the group of automorphisms of τn(X, x0). This homomorphism is used to obtain a necessary condition for the embedding of one transformation group in another, in particular, for the embedding of a discrete flow in a continuous flow with the same phase space.