Conjugations of unitary operators, II

For a unitary operator U on a separable complex Hilbert space H, we describe the set Cc(U) of all conjugations C (antilinear, isometric, and involutive maps) on H for which CUC=U. As this set might be empty, we also show that Cc(U)≠∅ if and only if U is unitarily equivalent to U∗.