Let $S$ be a quasi-projective scheme over base field $k$ and $X, Y$ two finite etale schemes over $S$ and assume we are in situation we know that the isom space $\text{Isom}_S(X,Y)$ exists as an $S$-scheme.

Concern: I'm wondering under which "reasonable/ natural" (= not too exotic) conditions this isom scheme $\text{Isom}_S(X,Y)$ is etale over $S$.

The motivation is coined by these explanations in the comments below the answer, where the focus lied on etaleness of $\text{Isom}_{\mathbb{G}_m}(E[\ell], (\mathbb{Z}/\ell\mathbb{Z})^2)$ where the $\ell$ torsion $E[\ell]$ of an elliptic curve $E/\mathbb{G}_m$ is assumed to be etale over $ \mathbb{G}_m$.

And this encourages my curiosity firstly to check which techniques are involved in the last statement's proof and how far it can be generalized. Ie is it something special for torsion groups of elliptic curves, or broader that the involved objects are group schemes or the rationality of the considered base $\mathbb{G}_m$?
Or can the etaleness of isom scheme above broadly (how far?) generalized dropping several properties the objects in the linked thread share?