  We construct a long exact sequence computing the obstruction space,
pi_1(BrPic(C_0)), to G-graded extensions of a fusion category C_0. The other
terms in the sequence can be computed directly from the fusion ring of C_0. We
apply our result to several examples coming from small index subfactors,
thereby constructing several new fusion categories as G-extensions. The most
striking of these is a Z/2Z-extension of one of the Asaeda-Haagerup fusion
categories, which is one of only two known 3-supertransitive fusion categories
outside the ADE series.
  In another direction, we show that our long exact sequence appears in exactly
the way one expects: it is part of a long exact sequence of homotopy groups
associated to a naturally occuring fibration. This motivates our constructions,
and gives another example of the increasing interplay between fusion categories
and algebraic topology.
