  Consider a coaction $\delta$ of a locally compact group $G$ on a \cstar
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 $A\times_\delta G\times_{\deltahat,r}N$ implements
equivalences between Mansfield induction of representations from $A\times G/N$
to $A\times G$ and restriction of representations from $A\times G\times_r N$ to
$A\times G$, and between restriction of representations from $A\times G$ to
$A\times G/N$ and Green induction of representations from $A\times G$ to
$A\times G\times_r N$. This allows us to deduce properties of Mansfield
induction from the known theory of ordinary crossed products.
