C(X) related to its topological homeomorphism group

When the space C(X) of continuous real-valued functions on X has the uniform topology, the space H(C(X)) of homeomorphisms on C(X) is a topological group when it has the fine topology. This article shows that for certain subgroups F and G of H(C(X)) and H(C(Y)), respectively, there is a natural one-to-one correspondence between a certain set of topological isomorphisms from F onto G and a certain set of homeomorphisms from C(X) onto C(Y) that relate to F and G. A number of examples are given of types of subgroups of H(C(X)) that satisfy this Correspondence Theorem and a weaker version of it.