  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 $D \subseteq
B \subseteq A$, 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
$\Sigma \rightarrow G$ is the twist associated with the embedding $D \subseteq
A$.
