  Massive elliptical galaxies, that serve as lenses in gravitational lensing
time delay measurements of the Hubble parameter $H_0$, often reside in a host
group. We consider degeneracies in the modeling of the group halo. When the
group effect on imaging can be summarized by its flexion (the next order term
beyond shear in the tidal expansion), the posterior likelihood map can develop
disjoint local minima, associated with an approximate discrete symmetry of a
dominant flexion term. Monte-Carlo Markov Chain (MCMC) algorithms that are not
designed to explore a rich posterior landscape can miss some of the minima,
introducing systematic bias. We study mock data and demonstrate that the bias
in $H_0$ can exceed $10\%$, and pulls the inference value of $H_0$ above its
truth value, for a reason that can be traced to the structure of a mismodeled
flexion term. MCMC algorithms that are designed to cope with a rich posterior
landscape can uncover the structure. If the group is X-ray bright enough, X-ray
data may also help to resolve the degeneracy, by pinpointing the group's center
of mass. Finally, we show that some implementations in the literature used an
inaccurate kinematical prior, mis-modeling the group velocity dispersion by as
much as $20\%$
