Abstract: In this talk, I will consider the question of whether there exist isomorphism-invariant versions of various classical results in combinatorial group theory. For example, the Higman-Neumann-Neumann Embedding Theorem states that any countable group G can be embedded into a 2-generator group K. In the standard proof of this classical theorem, the construction of the group K involves an enumeration of a set of generators of the group G; and it is clear that the isomorphism type of K usually depends upon both the generating set and the particular enumeration that is used.Consequently, it is natural to ask whether there exists a more uniform construction with the property that isomorphic groups G are embedded into isomorphic 2-generator groups K.