Following the notations and notions developped in Katz and Mazur's book "The arithmetic moduli of elliptic curves", we denote $\mathbf{(Ell)}$ the category whose objects are elliptic curves $E/S$ over variable base-schemes, and whose morphisms are Cartesian squares. A moduli problem is then simply a contravariant functor $\mathbf{(Ell)}\rightarrow \mathbf{(Set)}$. We say that a moduli problem $\mathcal{P}$ is étale if for any elliptic curve $E/S$, the functor $\mathbf{(Sch/S)}\rightarrow \mathbf{(Set)}$ sending an $S$-scheme $T$ to $\mathcal{P}(E_T/T)$ is representable by an étale $S$-scheme $\mathcal{P}_{E/S}$.
Let $N\geq 1$ be an integer and let $\Gamma(N)$ denote the "naive" full level $N$ moduli problem, which takes an elliptic curve $E/S$ and sends it to the set of $S$-group-schemes isomorphisms $$\phi:\left(\mathbb{Z}/N\mathbb{Z}\right)_S^2\xrightarrow{\sim} E[N]$$ Question: Is $\Gamma(N)$ étale over $\mathbf{(Ell)}$?
Actually, I know that it is relatively representable (and even representable by an elliptic curve over a smooth affine curve over $\mathbb{Z}[1/N]$, usually denoted $Y(N)$). If $E/S$ is any elliptic curve, then the $S$-scheme $\mathcal{P}_{E/S}$ associated to $\Gamma(N)$ is finite over $S$, and étale when $N$ is invertible on $S$. Is it also the case when $N$ is not invertible on $S$?
In other words, I know that it is étale when seen as a moduli problem on the category $\mathbf{(Ell_{\mathbb{Z}[1/N]})}$ of elliptic curves where $N$ is invertible, but is it also étale over the whole of $\mathbf{(Ell)}$?
NB: For context, this is claimed in Katz and Mazur's book on page 110 (page 61 in the pdf), and it is used on page 121 (page 66 in the pdf) when they describe its representing object $\mathbb{E}/Y(N)$ as a modular family.
I thank you very much for you clarifications.