Non-trivially graded self-dual fusion categories of rank $4$

Abstract: Let $\mathcal{C}$ be a self-dual spherical fusion categories of rank $4$ with non-trivial grading. We complete the classification of Grothendieck ring $K(\mathcal{C})$ of $\mathcal{C}$; that is, we prove that $K(\mathcal{C})\cong Fib\otimes\mathbb{Z}[\mathbb{Z}2]$, where $Fib$ is the Fibonacci fusion ring and $\mathbb{Z}[\mathbb{Z}_2]$ is the group ring on $\mathbb{Z}_2$. In particular, if $\mathcal{C}$ is braided then it is equivalent to $\textbf{Fib}\boxtimes\textbf{Vec}{\mathbb{Z}2}^{\omega}$ as fusion categories, where $\textbf{Fib}$ is a Fibonacci category and $\textbf{Vec}{\mathbb{Z}_2}^{\omega}$ is a rank $2$ pointed fusion category.