Graded extensions of generalized Haagerup categories

$\def\Z{\mathbb{Z}}$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 $\Z_{2n}$ generalized Haagerup categories for all $n \leq 5$; $\Z_2 \times \Z_2$-graded extensions of the Asaeda-Haagerup categories; and extensions of the $\Z_2 \times \Z_2$ generalized Haagerup category by its outer automorphism group $A_4$. The construction uses endomorphism categories of operator algebras, and in particular, free products of Cuntz algebras with free group $\mathrm{C}^\ast$-algebras.