Tetragonal intermediate modular curves

For every group $\{\pm 1\}\subseteq \Delta \subseteq (\mathbb {Z}/N\mathbb {Z})^\times $, there exists an intermediate modular curve $X_\Delta (N)$. We determine all curves $X_\Delta (N)$ whose $\mathbb {Q}$-gonality is equal to $4$, all curves $X_\Delta (N)$ whose $\mathbb {C}$-gonality is equal to $4$, and all curves $X_\Delta (N)$ whose $\mathbb {Q}$-gonality is equal to $5$. We also determine the $\mathbb {Q}$-gonality of all curves $X_\Delta (N)$ for $N\leq 40$ and $\{\pm 1\}\subsetneq \Delta \subsetneq (\mathbb {Z}/N\mathbb {Z})^\times $.