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 $3^{\mathbb{Z}\mathrm{/}2\mathbb{Z}\times \mathbb{Z}\mathrm{/}2\mathbb{Z}}$ 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 $\mathbb{Z}\mathrm{/}3\mathbb{Z}$-equivariantization ${\mathcal{C}}^{\mathbb{Z}\mathrm{/}3\mathbb{Z}}$ (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 ${\mathcal{C}}^{\mathbb{Z}\mathrm{/}3\mathbb{Z}}$, which are all $\mathbb{Z}\mathrm{/}3\mathbb{Z}$-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 $3^{\mathbb{Z}\mathrm{/}4\mathbb{Z}}$ subfactor; each admits $7$ simple module categories; and the Brauer-Picard group is $\mathbb{Z}\mathrm{/}2\mathbb{Z}$.