  We apply a method inspired by Popa's intertwining-by-bimodules technique to
investigate inner conjugacy of MASAs in graph $C^*$-algebras. First we give a
new proof of non-inner conjugacy of the diagonal MASA ${\mathcal D}_n$ to its
non-trivial image under a quasi-free automorphism, where $E$ is a finite
transitive graph. Changing graphs representing the algebras, this result
applies to some non quasi-free automorphisms as well. Then we exhibit a large
class of MASAs in the Cuntz algebra ${\mathcal O}_n$ that are not inner
conjugate to the diagonal ${\mathcal D}_n$.
