  We consider the following question: Let $\mathcal{A}$ be an abelian
self-adjoint algebra of bounded operators on a Hilbert space $\mathcal{H}$.
Assume that $\mathcal{A}$ is invariant under conjugation by a unitary operator
$U$, i.e., $U^* AU$ is in $\mathcal{A}$ for every member $A$ of $\mathcal{A}$.
Is there a maximal abelian self-adjoint algebra containing $\mathcal{A}$, which
is still invariant under conjugation by $U$? The answer, which is easily seen
to be yes in finite dimensions, is not trivial in general. We prove affirmative
answers in special cases including the one where $\mathcal{A}$ is generated by
a compact operator. We also construct a counterexample in the general case,
whose existence is perhaps surprising.
