  Time-delay cosmography of lensed quasars has achieved 2.4% precision on the
measurement of the Hubble constant, $H_0$. As part of an ongoing effort to
uncover and control systematic uncertainties, we investigate three potential
sources: 1- stellar kinematics, 2- line-of-sight effects, and 3- the deflector
mass model. To meet this goal in a quantitative way, we reproduced the
H0LiCOW/SHARP/STRIDES (hereafter TDCOSMO) procedures on a set of real and
simulated data, and we find the following. First, stellar kinematics cannot be
a dominant source of error or bias since we find that a systematic change of
10% of measured velocity dispersion leads to only a 0.7% shift on $H_0$ from
the seven lenses analyzed by TDCOSMO. Second, we find no bias to arise from
incorrect estimation of the line-of-sight effects. Third, we show that
elliptical composite (stars + dark matter halo), power-law, and cored power-law
mass profiles have the flexibility to yield a broad range in $H_0$ values.
However, the TDCOSMO procedures that model the data with both composite and
power-law mass profiles are informative. If the models agree, as we observe in
real systems owing to the "bulge-halo" conspiracy, $H_0$ is recovered precisely
and accurately by both models. If the two models disagree, as in the case of
some pathological models illustrated here, the TDCOSMO procedure either
discriminates between them through the goodness of fit, or it accounts for the
discrepancy in the final error bars provided by the analysis. This conclusion
is consistent with a reanalysis of six of the TDCOSMO (real) lenses: the
composite model yields $74.0^{+1.7}_{-1.8}$ $km.s^{-1}.Mpc^{-1}$, while the
power-law model yields $H_0=74.2^{+1.6}_{-1.6}$ $km.s^{-1}.Mpc^{-1}$. In
conclusion, we find no evidence of bias or errors larger than the current
statistical uncertainties reported by TDCOSMO.

Cosmology and Nongalactic Astrophysics