Links in the spherical 3-manifold obtained from the quaternion group and their lifts

We show that there are infinitely many triples of non-isotopic hyperbolic links in the lens space $L(4,1)$ such that the three lifts of each triple in $S^3$ are isotopic. They are obtained as the lifts of links in $S^3\mathrm{/}{Q}_8$ by double covers, where $Q_8$ is the quaternion group. To construct specific examples, we introduce a diagram of a link in $S^3\mathrm{/}{Q}_8$ obtained by projecting to a square. The diagrams of isotopic links are connected by Reidemeister-type moves.