Strongly lensed quasars can provide measurements of the Hubble constant ($H_{0}$) independent of any other methods. One of the key ingredients is exquisite high-resolution imaging data, such as Hubble Space Telescope (HST) imaging and adaptive-optics (AO) imaging from ground-based telescopes, which provide strong constraints on the mass distribution of the lensing galaxy. In this work, we expand on the previous analysis of three time-delay lenses with AO imaging (RXJ1131-1231, HE0435-1223, and PG1115+080), and perform a joint analysis of J0924+0219 by using AO imaging from the Keck Telescope, obtained as part of the SHARP (Strong lensing at High Angular Resolution Program) AO effort, with HST imaging to constrain the mass distribution of the lensing galaxy. Under the assumption of a flat $Lambda$CDM model with fixed $Omega_{rm m}=0.3$, we show that by marginalizing over two different kinds of mass models (power-law and composite models) and their transformed mass profiles via a mass-sheet transformation, we obtain $Delta t_{rm BA}hhat{sigma}_{v}^{-2}=6.89substack{+0.8-0.7}$ days, $Delta t_{rm CA}hhat{sigma}_{v}^{-2}=10.7substack{+1.6-1.2}$ days, and $Delta t_{rm DA}hhat{sigma}_{v}^{-2}=7.70substack{+1.0-0.9}$ days, where $h=H_{0}/100~rm km,s^{-1},Mpc^{-1}$ is the dimensionless Hubble constant and $hat{sigma}_{v}=sigma^{rm ob}_{v}/(280~rm km,s^{-1})$ is the scaled dimensionless velocity dispersion. Future measurements of time delays with 10% uncertainty and velocity dispersion with 5% uncertainty would yield a $H_0$ constraint of $sim15$% precision.
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