On the realization of a class of $\text{SL}(2,\mathbb{Z})$-representations

Let $p<q$ be odd primes, ${\rho }_1$ and ${\rho }_2$ be irreducible representations of $\text{SL}(2,{\mathbb{Z}}_p)$ and $\text{SL}(2,{\mathbb{Z}}_q)$ of dimensions $\frac{p+1}{2}$ and $\frac{q+1}{2}$, respectively. We show that if ${\rho }_1\oplus {\rho }_2$ can be realized as modular representation associated to a modular fusion category $\mathcal{C}$, then $q-p=4$. Moreover, if $\mathcal{C}$ contains a non-trivial étale algebra, then $\mathcal{C}⊠\mathcal{C}({\mathbb{Z}}_p,\eta )\cong \mathcal{Z}(\mathcal{A})$ as braided fusion category, where $\mathcal{A}$ is a near-group fusion category of type $({\mathbb{Z}}_p,p)$. And we show that there exists a non-trivial ${\mathbb{Z}}_2$-extension of $\mathcal{A}$ that contains simple objects of Frobenius-Perron dimension $\frac{\sqrt{p}+\sqrt{q}}{2}$.