I have some questions about what feels like basic topics in quotients of schemes by group actions. Consequently I suspect there are well-known references; I couldn't find them, though.

First definition: isovariant morphisms

Throughout, I fix a finite group $G$. For a $G$-scheme $X$ we have the stabilizer scheme $Stab(X) \subset G \times X$, which may be informally described as the scheme of pairs $(g, x)$ with $gx=x$. In other words the fibers of $Stab(X) \to X$ are given by the scheme-theoretic stabilizers of the points. If $f: X \to Y$ is a morphism of $G$-schemes, there is a natural map $Stab(X) \to Stab(Y) \times_Y X =: f^* Stab(Y)$. We call $f$ isovariant (a.k.a. stabilizer preserving, fixed point reflecting) if $Stab(X) \to f^* Stab(Y)$ is an isomorphism.

Preliminary question

If $f: X \to Y$ is isovariant, then for every $x \in X$ the stabilizer of $G$ at $x$ is equal to the stabilizer of $G$ at $f(x)$.

Question: are there conditions for the converse to hold?

If I interpret the stacks project correctly, the converse does hold if $f$ is unramified. Can we do better?

More definitions: strong morphisms

From now on, I fix a $G$-scheme $S$ and assume that the quotient $S/G$ exists (in algebraic spaces, if we want to be fancy). For safety, assume that the quotient is universal (e.g. $|G|$ invertible). Let $P$ be some property of morphisms of schemes, such as flat, smooth, etc. A morphism $f: X \to S$ is called strongly $P$ if $X/G$ exists, $f/G$ is $P$, and $X = X/G \times_{S/G} S$.

Note that if $f': X' \to S/G$ is any morphism, then $f = f' \times_{S/G} S$ is isovariant. Morphisms that arise in this way are called strongly isovariant.

Main questions

Basically I want to know: if $f: X \to S$ is $P$ and isovariant, then under what additional hypotheses is $f$ strongly $P$?

I'm particularly interested in the case $P = \emptyset$ (which isovariant morphisms are strongly isovariant), $P$ means "flat" and $P$ means "smooth".

Comments

It is easy to see that this is true (possibly under mild assumptions) if $P$ means "étale". Also if I'm not mistaken, smooth + strongly flat implies strongly smooth.