Is there a subfactor construction involving 2-groups?
I seem to recall that there is a straightforward subfactor construction that yields fusion categories given by G-graded vector spaces and representations of G, for finite groups G. Is there an analogous construction for 2-groups?
Some background: A 2-group is a monoidal groupoid, for which the isomorphism classes of objects form a group. Sinh showed that up to monoidal equivalence, these are classified by a group G (isom. classes of objects), a G-module H (automorphisms of identity), and an element of H3(G,H). In the context of this discussion, we can limit our attention to G finite, H=Cx. One notable feature is that when the action of G on H is trivial, the three-cocycle twists the associator in the G-graded vector space category.
I'm mostly curious about how to tell when two elements of H3(G,H) yield Morita-equivalent fusion categories, and am wondering if subfactors or planar algebras make it easy to detect this.