Tetragonal modular quotients $X_0^{+d}(N)$

Let $N$ be a positive integer. For every $d\mathrm{\mid }N$ such that $(d,N\mathrm{/}d)=1$ there exists an Atkin-Lehner involution $w_d$ of the modular curve $X_0(N)$. In this paper we determine all quotient curves $X_0(N)\mathrm{/}{w}_d$ whose $\mathbb{Q}$-gonality is equal to $4$ and all quotient curves $X_0(N)\mathrm{/}{w}_d$ whose $\mathbb{C}$-gonality is equal to $4$.