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Let A be a C*-algebra and let D be a Cartan subalgebra of A. We study the following question: if B is a C*-algebra such that D B A, is D a Cartan subalgebra of B? We give a positive answer in two cases: the case when there is a faithful conditional expectation from A onto B, and the case when A is nuclear and D is a C*-diagonal of A. In both cases there is a one-to-one correspondence between the intermediate C*-algebras B, and a class of open subgroupoids of the groupoid G, where ! G is the twist associated with the embedding D A.