Let $G(N)$ be the congruence subgroup

$\big\{ \begin{pmatrix} a&b \\ c&d \end{pmatrix} \in SL_2(\mathbb{Z}) \ \ | \ \ a \equiv d \mod N \textrm{ and } b \equiv c\equiv 0 \mod N \big\}$.

$G(N) \backslash \mathbb{H}$ seems to parametrize elliptic curves with a cyclic subgroup and some extra data (I think its automorphism group is $PSL_2(\mathbb{Z} / N \mathbb{Z})$, intuitively it is moving cyclic subgroups but fixing their Weil pairings).

Has this quotient been considered already somewhere in the literature? Ideally I would like to know exactly what it is the moduli space for, and how it relates to $\Gamma_0(N) \backslash \mathbb{H}$.