Zariski's main theorem in the form of Grothendieck [EGA 4-3, Thm. 8.12.6] asserts that any quasi-finite morphism $a : V^{\circ} → W$ factors as the composition of an open immersion and a finite morphism, say $$V^{\circ} \xrightarrow{\alpha\text{, open immersion}} V \xrightarrow{\beta\text{, finite}} W.$$

I believe that if $V^{\circ}$ is normal, this factorisation satisfies a number of universal properties. Below, I list three properties which I believe hold true at least for varieties defined over $\mathbb C$. All this must be known to experts, but I was not able not find anything in the literature. I would like to ask if anyone could point me to a reference for these matters.

(1) The intermediate variety $V$ can be chosen to be normal. With this choice, the variety $V$ and the factorisation is unique up to unique isomorphism.

(2) If $G$ is any group which acts algebraically on $V°$ and $W$ and if $a$ is equivariant with respect to these actions, then $G$ also acts on $V$ and the morphisms $α$ and $β$ are equivariant with respect to these actions.

(3) If $W° := \mathrm{Image}(a)$ is open in $W$ and $a : V° \to W°$ is Galois with group $G$, then $β: V → W$ is Galois with group $G$.