Bielliptic intermediate modular curves

We determine which of the modular curves $X_{\mathrm{\Delta }}(N)$, that is, curves lying between $X_0(N)$ and $X_1(N)$, are bielliptic. Somewhat surprisingly, we find that one of these curves has exceptional automorphisms. Finally we find all $X_{\mathrm{\Delta }}(N)$ that have infinitely many quadratic points over $\mathbb{Q}$.