  The properties of low-mass dark matter (DM) halos appear to be remarkably
diverse relative to cold, collisionless DM predictions, even in the presence of
baryons. We show that self-interacting DM (SIDM) can simultaneously explain
observations of halo diversity at two opposite extremes$-$the inner density
profile of the dense substructure perturbing the strong lens galaxy
SDSSJ0946+1006 and the rotation curves of isolated, gas-rich ultradiffuse
galaxies (UDGs). To achieve this, we present the first cosmological zoom-in
simulation featuring strong DM self-interactions in a galaxy group environment
centered on a $10^{13}~M_{\mathrm{\odot}}$ host halo. In our SIDM simulation,
most surviving subhalos of the group-mass host are deeply core-collapsed,
yielding excellent candidates for the observed dense strong-lens perturber.
Self-interactions simultaneously create kiloparsec-scale cores in
low-concentration isolated halos, which could host the observed UDGs. Our
scenario can be further tested with observations of DM structure and galaxies
over a wide mass range.
