Isometries between groups of invertible elements in Fourier-Stieltjes algebras

We prove that if open subgroups of the groups of invertible elements in two Fourier-Stieltjes algebras are isometric as metric spaces, then the underlying locally compact groups are topologically isomorphic. We describe the structure of isometric real algebra isomorphisms between Fourier-Stieltjes algebras and apply it to prove the above result.