  We consider quotients of mapping class groups of orientable, finite type
surfaces by subgroups whose action on the curve graph has large displacement.
This class includes quotients by the normal closure of a pseudo-Anosov element,
the mapping class group itself, and in view of forthcoming work of Abbott,
Berlyne, Ng, and Rasmussen, also random quotients. First, we show that every
automorphism of the corresponding quotient of the curve graph is induced by a
mapping class, thus generalising Ivanov's Theorem about automorphisms of the
curve graph. Then we use this to prove quasi-isometric rigidity under
additional assumptions, satisfied by all aforementioned quotients. In the
process, we clarify a proof of quasi-isometric rigidity of mapping class groups
by Behrstock, Hagen, and Sisto. Finally, we show that the outer automorphisms
groups of our quotients, as well as their abstract commensurators, are "the
smallest possible".
