Conjugations of Unitary Operators, II

For a given unitary operator $U$ on a separable complex Hilbert space $\text{h}$, we describe the set ${\mathcal{C}}_c(U)$ of all conjugations $C$ (antilinear, isometric, and involutive maps) on $\text{h}$ for which $CUC=U$. As this set might be empty, we also show that ${\mathcal{C}}_c(U)≠\mathrm{\varnothing }$ if and only if $U$ is unitarily equivalent to $U^{\ast }$.