Let $k\geq 0$, $N\geq 2$ be integers. I suspect that there is an action of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$ on the finite-dimensional $\mathbb{Q}$-vector space of holomorphic modular forms of level $\Gamma(N)$ and weight $k$ such that the invariant subspace is the space of holomorphic modular forms of level $\Gamma(1)$ and weight $k$ (considered as level $\Gamma(N)$ modular forms).
I think so because in the moduli interpretation of the modular curve $X(N)$ the group $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$ acts on the set of pairs of generators of the $N$-torsion group on a generalized elliptic curve (but for $N=2$ the curve is stacky so I am not sure).
First question: does such an action indeed exist and where can I read more about it? Second question: can I explicitly write down what this action does to the Fourier coefficients of modular forms?