The symmetric imprimitivity theorem provides a Morita equivalence between two crossed products of induced C*-algebras and includes as special cases many other important Morita equivalences such as Green's imprimitivity theorem. We show that the symmetric imprimitivity theorem is compatible with various inflated actions and coactions on the crossed products.
Echterhoff, S. & Raeburn, I. F. (2000). Induced C*-algebras, coactions and equivariance in the symmetric imprimitivity theorem. Mathematical Proceedings of the Cambridge Philosophical Society, 128 (2), 327-342.