Tetragonal intermediate modular curves

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 $X_{\mathrm{\Delta }}(N)$ whose $\mathbb{Q}$-gonality is equal to $4$, all curves $X_{\mathrm{\Delta }}(N)$ whose $\mathbb{C}$-gonality is equal to $4$, and all curves $X_{\mathrm{\Delta }}(N)$ whose $\mathbb{Q}$-gonality is equal to $5$. We also determine the $\mathbb{Q}$-gonality of all curves $X_{\mathrm{\Delta }}(N)$ for $N\le 40$ and $\{\pm 1\}⊊\mathrm{\Delta }⊊(\mathbb{Z}\mathrm{/}N\mathbb{Z}{)}^{\times }$.