Duality of restriction and induction for C*-coactions

Consider a coaction δ of a locally compact group G on a C*-algebra A, and a closed normal subgroup N of G. We prove, following results of Echterhoff for abelian G, that Mansfield's imprimitivity between A x$| G/N and A × f G × δ r N implements equivalences between Mansfield induction of representations from A × δ| G/N to A x& G and restriction of representations from A× iG× frNtoA× (G, and between restriction of representations from A × δ G to A × δ| G/N and Green induction of representations from A × g G to A × f G × j r N. This allows us to deduce properties of Mansfield induction from the known theory of ordinary crossed products.