Bielliptic quotient modular curves of $X_0(N)$

Let $N\ge 1$ be a non-square free integer and let $W_N$ be a non-trivial subgroup of the group of the Atkin-Lehner involutions of $X_0(N)$ such that the modular curve $X_0(N)\mathrm{/}{W}_N$ has genus at least two. We determine all pairs $(N,W_N)$ such that $X_0(N)\mathrm{/}{W}_N$ is a bielliptic curve and the pairs $(N,W_N)$ such that $X_0(N)\mathrm{/}{W}_N$ has an infinite number of quadratic points over $\mathbb{Q}$.