Conjugations of unitary operators, II

For a unitary operator U on a separable complex Hilbert space \({\mathcal {H}}\), we describe the set \({\mathscr {C}}_{c}(U)\) of all conjugations C (antilinear, isometric, and involutive maps) on \({\mathcal {H}}\) for which \(C U C = U\). As this set might be empty, we also show that \({\mathscr {C}}_{c}(U) \not = \varnothing \) if and only if U is unitarily equivalent to \(U^{*}\).