  There are at least three imprimitivity bimodules naturally associated to a
maximal coaction of a discrete group G on a C*-algebra and a normal subgroup of
G: Mansfield's bimodule; the bimodule assembled by Ng from Green's
imprimitivity bimodule and Katayama duality; and a bimodule assembled from
Green's bimodule and a crossed-product Mansfield bimodule. We show that all
three of these are isomorphic, so that the corresponding inducing maps on
representations are identical. This can be interpreted as saying that Mansfield
and Green induction are inverses of one another ``modulo Katayama duality''.
These results pass to twisted coactions; dual results starting with an action
are also given.
