  Let $K$ be a field, let $G$ be a finite group, and let $\bar X\rightarrow
\bar Y$ be a $G$-Galois branched cover of varieties over $K^{sep}$. Given a
mere cover model $X\rightarrow Y$ of this cover over $K$, in Part I of this
paper I observe that there is a unique minimal field $E$ over which
$X\rightarrow Y$ becomes Galois, and I prove that $E/K$ is Galois with group a
subgroup of $Aut(G)$. In Part II of this paper, by making the additional
assumption that $K$ is a field of definition (i.e., that there exists \it some
\rm Galois model over $K$), I am able to give an explicit description of the
unique minimal field of Galois action for $X\rightarrow Y$. Namely, if there
exists a $K$-rational point of $X$ above an unramified point $P\in Y(K)$ then
$E$ is contained in the intersection of the specializations at $P$ in the
various different $G$-Galois models of $\bar X\rightarrow \bar Y$ over $K$.
Using the same proof mechanism, I observe a reverse version of "The Twisting
Lemma", which asserts that the behavior of the $K$-rational points on the
various mere cover models over $K$, and the behavior of the specializations on
the various $G$-Galois models over $K$, are all governed by a single
equivalence relation (independent of the model) on the $K$-rational points of
the base variety.
