  SDSS J2222+2745, at z = 0.489, is one of the few currently known lens
clusters with multiple images of a background (z = 2.801) quasar with measured
time delays. We combine imaging from the Hubble Space Telescope (HST) with
recent Multi Unit Spectroscopic Explorer (MUSE) spectroscopic data to securely
identify 34 cluster members and 12 multiple images from 3 background sources.
We measure the stellar velocity dispersions of 13 cluster galaxies, enabling an
independent estimate of the contribution of the sub-halo mass component to the
lens total mass. The projected total mass distribution of the lens cluster is
best modelled with a single large-scale mass component, a galaxy-scale
component, anchored by the MUSE kinematic information, and an external shear.
The best-fit strong lensing model yields a root mean square separation between
the model-predicted and observed positions of the multiple images of 0".29.
When analysing the impact of systematic uncertainties, stemming from modelling
assumptions and used observables, we find that the projected total mass
profile, relative weight of the sub-halo mass component, and critical lines are
consistent, within the statistical uncertainties. The predicted magnification
and time delay values are, instead, more sensitive to the local details of the
lens total mass distribution, and vary significantly among lens models that are
similarly good at reproducing the observed multiple image positions. Due to its
complex morphology, the low number of point-like multiple images, and current
model degeneracies, it becomes clear that additional information (from the
observed surface brightness distribution of lensed sources and the measured
time delays) needs to be included in the modelling of SDSS J2222+2745 for
accurate and precise cosmological measurements. The full MUSE secure
spectroscopic catalogue presented in this work is made publicly available.

Astrophysics of Galaxies;Cosmology and Nongalactic Astrophysics