It is known that a completely bounded algebra homomorphism φ : A(G) → B(H) is given by a piecewise affine map α : Y ⊂ H → G, when G is amenable and H a locally compact group ( P.J. Cohen, B. Host, M. Ilie , N. Spronk ). Then φ has a canonical extension to the Fourier Stieltjes algebra of B(G) given by α. We study the connection between arbitrary extensions and the canonical one, in particular uniqueness properties. As an application we obtain description of completely bounded homomorphisms of the Fourier Stieltjes algebras, using along the way a characterization of w ∗ -continuous homomorphisms given by piecewise affine maps. This is joint work with R. Stokke.