  We show that many normal subgroups of the braid group modulo its centre, and
of the mapping class group of a sphere with marked points, have the property
that their automorphism and abstract commensurator groups are mapping class
groups of such spheres. As one application, we establish the automorphism
groups of each term in the lower central series and derived series of the pure
braid group. We also obtain new proofs of results of Dyer-Grossman and Orevkov,
showing that the automorphism groups of the braid group and of its commutator
subgroup are isomorphic. We then calculate the automorphism and abstract
commensurator groups of each term in the hyperelliptic Johnson filtration,
recovering a result of Childers for the Torelli case. The techniques used in
the paper rely on resolving a metaconjecture of Nikolai V. Ivanov for "graphs
of regions", extending work of Brendle-Margalit.
