Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when α is a group homomorphism which pushes forward the Haar measure of G to a measure absolutely continuous with respect to the Haar measure on H, then (α×α)−1 preserves sets of compact operator synthesis, and conversely when α is onto. We also prove similar preservation results for operator Ditkin sets and operator M-sets, obtaining preservation results for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov and Turowska.