Tetragonal modular quotients of $X_0(N)$

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 $X_0(N)$. Let $B(N)$ be the group of all such involutions. In this paper we determine all $\mathbb{C}$ and $\mathbb{Q}$-tetragonal quotient curves $X_0(N)\mathrm{/}{W}_N$, where $W_N\subseteq B(N)$ such that $4\le \mathrm{\mid }{W}_N\mathrm{\mid }\le 2^{\omega (N)-1}$, thus completing the classification of all $\mathbb{C}$-tetragonal quotients of $X_0(N)$ by Atkin-Lehner involutions.