Duality of restriction and induction for $C^*$-coactions

Consider a coaction $\delta$ 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 $A\times _{\delta |}G/N$ and $A\times _\delta G\times _{\hat {\delta } ,r}N$ implements equivalences between Mansfield induction of representations from $A\times _{\delta |}G/N$ to $A\times _\delta G$ and restriction of representations from $A\times _\delta G\times _{\hat {\delta } ,r}N$ to $A\times _\delta G$, and between restriction of representations from $A\times _\delta G$ to $A\times _{\delta |}G/N$ and Green induction of representations from $A\times _\delta G$ to $A\times _\delta G\times _{\hat {\delta } ,r}N$. This allows us to deduce properties of Mansfield induction from the known theory of ordinary crossed products.