Factorization and extension of isomorphisms on C(X) to homeomorphisms on hyperspaces

For a binormal space X, the space C(X) of continuous real-valued functions on X under the graph topology is a topological group that is a subspace of a certain hyperspace L(X) under the Vietoris topology. Conditions are given showing when a positive topological group isomorphism from C(X) onto C(Y) has an extension to an ordered homeomorphism from L(X) onto L(Y). It is then shown how such an ordered homeomorphism can be uniquely factored through a homeomorphism from X onto Y. Finally, the existence of such an extension and the existence of such a factorization (restricted to C(X)) are shown to be equivalent.