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Massive elliptical galaxies, that serve as lenses in gravitational lensingtime delay measurements of the Hubble parameter $H_0$, often reside in a hostgroup. We consider degeneracies in the modeling of the group halo. When thegroup effect on imaging can be summarized by its flexion (the next order termbeyond shear in the tidal expansion), the posterior likelihood map can developdisjoint local minima, associated with an approximate discrete symmetry of adominant flexion term. Monte-Carlo Markov Chain (MCMC) algorithms that are notdesigned to explore a rich posterior landscape can miss some of the minima,introducing systematic bias. We study mock data and demonstrate that the biasin $H_0$ can exceed $10\%$, and pulls the inference value of $H_0$ above itstruth value, for a reason that can be traced to the structure of a mismodeledflexion term. MCMC algorithms that are designed to cope with a rich posteriorlandscape can uncover the structure. If the group is X-ray bright enough, X-raydata may also help to resolve the degeneracy, by pinpointing the group's centerof mass. Finally, we show that some implementations in the literature used aninaccurate kinematical prior, mis-modeling the group velocity dispersion by asmuch as $20\%$

Host group degeneracy in gravitational lensing time delay determination of $H_0$