Tetragonal modular quotients $X_0^{\ast }(N)$

Let $N$ be a positive integer. For every $d\mid N$ such that $(d,N\mathrm{/}d)=1$ there exists an Atkin-Lehner involution $w_d$ of the modular curve $X_0(N)$. The curve $X_0^{\ast }(N)$ is a quotient curve of $X_0(N)$ by $B(N)$, the group of all involutions $w_d$. In this paper we determine all quotient curves $X_0^{\ast }(N)$ whose $\mathbb{C}$-gonality is equal to $4$. We also determine all curves $X_0^{\ast }(N)$ whose $\mathbb{Q}$-gonality is equal to $4$ with the exception of level $N=378$.