  We prove that if $\rho: A(H) \to B(G)$ is a homomorphism between the Fourier
algebra of a locally compact group $H$ and the Fourier-Stieltjes algebra of a
locally compact group $G$ induced by a mixed piecewise affine map $\alpha : G
\to H$, then $\rho$ extends to a w*-w* continuous map between the corresponding
$L^\infty$ algebras if and only if $\alpha$ is an open map. Using techniques
from TRO equivalence of masa bimodules we prove various transference results:
We show that when $\alpha$ is a group homomorphism which pushes forward the
Haar measure of $G$ to a measure absolutely continuous with respect to the Haar
measure of $H$, then $(\alpha\times\alpha)^{-1}$ preserves sets of compact
operator synthesis, and conversely when $\alpha$ is onto. We also prove similar
preservation properties for operator Ditkin sets and operator M-sets, obtaining
preservation properties for M-sets as corollaries. Some of these results extend
or complement existing results of Ludwig, Shulman, Todorov and Turowska.
