Induction in stages for crossed products of C^*-algebras by maximal coactions

Let δ be a maximal coaction of a locally compact group G on a C^*-algebra B, and let N and H be closed normal subgroups of G with N ⊆ H. We show that the process Ind_{G / H}^G which uses Mansfield's bimodule to induce representations of B ⋊_δ G from those of B ⋊_{δ |} (G / H) is equivalent to the two-stage induction process Ind_{G / N}^G ○ Ind_{G / H}^{G / N}. The proof involves a calculus of symmetric imprimitivity bimodules which relates the bimodule tensor product to the fibred product of the underlying spaces.