  As a fundamental parameter for modern cosmology, the Hubble constant $H_0$ is
experiencing a serious crisis. In this paper, we explore an independent
approach to measure $H_0$ based on the time-delay cosmography with strong
gravitational lensing of a quasar by a galaxy cluster. Specifically we focus on
the strong lensing system SDSS J1004+4112 with the maximum image separation of
14.62$''$, the first system of a quasar lensed by a galaxy cluster with five
multiple images. Incorporating the latest time-delay measurements, we
investigate the lens model dependence from the combination of 16 different lens
mass models. We find that the lens model dependence is indeed large, with the
combined measurement of the Hubble constant of
$H_0=67.5^{+14.5}_{-8.9}km/s/Mpc$ that is obtained by summing posteriors of the
Hubble constant from the 16 models with equal weighting. Interestingly, our
results show that the value of Hubble constant decreases as the complexity of
the perturbation around the lens increases, although weighting based on
positional errors of quasar images does not significantly improve the $H_0$
constraint. We find that the 16 different mass models predict largely different
shapes of the lensed quasar host galaxy as well as other lensed galaxies behind
the cluster. By selecting two mass models that best reproduces those shapes,
the constraint on the Hubble constant is significantly tightened to
$H_0=59.1^{+3.6}_{-3.5}km/s/Mpc$. While we caution that our analysis still does
not fully explore all the possible mass model uncertainty, our results
highlight the importance of including as many constraints as possible such as
extended shapes of lensed galaxies for obtaining tight constraints on the
Hubble constant from cluster-lensed quasar lens systems.
