  We prove that the Brauer-Picard group of Morita autoequiv- alences of each of
the three fusion categories which arise as an even part of the Asaeda-Haagerup
subfactor or of its index 2 extension is the Klein four-group. We describe the
36 bimodule categories which occur in the full subgroupoid of the Brauer-Picard
groupoid on these three fusion categories. We also classify all irreducible
subfactors both of whose even parts are among these categories, of which there
are 111 up to isomorphism of the planar algebra (76 up to duality). Although we
identify the entire Brauer-Picard group, there may be additional fusion
categories in the groupoid. We prove a partial classification of possible
additional fusion categories Morita equivalent to the Asaeda-Haagerup fusion
categories and make some conjectures about their existence; we hope to address
these conjectures in future work.
