Intermediate modular curves with infinitely many quartic points

For every group $\{\pm 1\}\subseteq \mathrm{\Delta }\subseteq (\mathbb{Z}\mathrm{/}N\mathbb{Z}{)}^{\times }$, there exists an intermediate modular curve $X_{\mathrm{\Delta }}(N)$. In this paper we determine all curves $X_{\mathrm{\Delta }}(N)$ with infinitely many points of degree $4$ over $\mathbb{Q}$. To do that, we developed a method to compute possible degrees of rational morphisms from $X_{\mathrm{\Delta }}(N)$ to an elliptic curve.