Graded extensions of generalized Haagerup categories

We classify certain Z(2)-graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including: Z(2)-graded extensions of Z(2)n generalized Haagerup categories for all n <= 5; Z(2) xZ(2)-graded extensions of the Asaeda-Haagerup categories; and extensions of the Z(2) x Z(2) generalized Haagerup category by its outer automorphism group A4. The construction uses endomorphism categories of operator algebras, and in particular, free products of Cuntz algebras with free group C*-algebras.