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 probably some of the properties enumerated in previous sentence the concrete objects in the linked thread share?

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

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

The motivation is coined by the comments below Daniel Litt's answer to Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?, where the focus lay on étaleness of $\operatorname{Isom}_{\mathbb{G}_m}(E[\ell], (\mathbb{Z}/\ell\mathbb{Z})^2)$ where the $\ell$ torsion $E[\ell]$ of an elliptic curve $E/\mathbb{G}_m$ is assumed to be étale 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. I.e., 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 étaleness of isom scheme above be broadly (how far?) generalized dropping probably some of the properties enumerated in the previous sentence the concrete objects in the linked thread share?

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?

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 probably some of the properties enumerated in previous sentence the concrete objects in the linked thread share?