Fusion Categories Associated to Subfactors with Index $3+\sqrt{5}$

We classify fusion categories which are Morita equivalent to even parts of subfactors with index $3+\sqrt{5}$, and module categories over these fusion categories. For the fusion category $\mathcal{C}$ which is the even part of the self-dual $3^{\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} }$ subfactor, we show that there are $30$ simple module categories over $\mathcal{C}$; there are no other fusion categories in the Morita equivalence class; and the order of the Brauer-Picard group is $360$. The proof proceeds indirectly by first describing the Brauer-Picard groupoid of a $\mathbb{Z}/3\mathbb{Z}$-equivariantization $\mathcal{C}^{\mathbb{Z}/3\mathbb{Z} }$ (which is the even part of the $4442$ subfactor). We show that that there are exactly three other fusion categories in the Morita equivalence class of $\mathcal{C}^{\mathbb{Z}/3\mathbb{Z} }$, which are all $\mathbb{Z}/3\mathbb{Z}$-graded extensions of $\mathcal{C}$. Each of these fusion categories admits $20$ simple module categories, and their Brauer-Picard group is $\mathcal{S}_3$. We also show that there are exactly five fusion categories in the Morita equivalence class of the even parts of the $3^{\mathbb{Z}/4\mathbb{Z} }$ subfactor; each admits $7$ simple module categories; and the Brauer-Picard group is $\mathbb{Z}/2\mathbb{Z}$.