Conjugations of Unitary Operators, II

For a given unitary operator U on a separable complex Hilbert space, we describe the set C₂ (U) of all conjugations C (antilinear, isometric, and involutive maps) on for which C U C = U. As this set might be empty, we also show that C₂ (U) = if and only if U is unitarily equivalent to U^*.