Homotopy-invariant group constructions and preimages of essential maps

As is well known, the suspension [epsilon]X of a space X has a Co-H group structure in terms of which the collection of all homotopy classes of base point preserving maps [epsilon]X -> Y can be shown to be a group. To put it briefly, [epsilon] is a covariant functor from the category of pointed topological spaces and base point preserving maps to the category of groups and homomorphisms. [...]