Etale fundamental group action on set-theoretic fiber Let $f: Y \to X$ be a finite etale cover of schemes. Fix a geometric point $x \in X$. I would like $\pi_1(X,x)^{et}$, the etale fundamental group, to act on the set-theoretic fiber of $x$. This set is the same as sets as the scheme-theoretic fiber of $x$. But we have an action only on the geometric fiber of $x$. Is there a way I can induce an action on the set-theoretic fiber of x? (Maybe it injects to the geometric fiber of $x$ through a set map?)
I'd appreciate any inputs on this. Thanks!
 A: I'm afraid that this won't be possible in general. As Alex Youcis already remarked, you get a surjective map from the geometric fiber into the set-theoretic fiber which comes from the functoriality of the fiber product. Now for connected $Y$, the action of the étale fundamental group on the geometric fiber is transitive, so the action cannot descend if e.g. the geometric fiber has three and the actual fiber has two points.
For a concrete (and very ad hoc, there should be better ones!) example of this happening you could look at the localization $R=\mathbb{Z}_{(5)}$ of $\mathbb{Z}$ on the prime ideal $(5)$ and consider
$$
Spec(R[X]/(X^3-2)) \to Spec(R)
$$
which is readily seen to be a connected étale cover ($X^3-2$ is irreducible over $\mathbb{Q}$, flatness bc. the leading coefficient is 1, finiteness is clear and unramifiedness can be seen at the fibers, see below). 
As $3^3 =2$ in $\mathbb{F}_5$ and the polynomial 
$$
X^2+3X+4 = (X^3-2):(X-3)
$$ 
has no zeros in $\mathbb{F}_5$,
we see that the set theoretic fiber over the closed point $(5)$ of $R$ is of the form
$$
Spec(\mathbb{F}_5) \coprod Spec(\mathbb{F}_{25})
$$
where the second point splits into two points in the geometric fiber. As mentioned above, the action of the étale fundamental group will be transitive on geometric points, so you can map the point over $Spec(\mathbb{F}_5)$ to one lying over the other; this can't be well-defined on the set-theoretic fiber.
We can be a bit more concrete about this action by fixing one of the noncanonical isomorphisms between the étale fundamental group in the generic point and the étale fundamental group in our closed point $(5)$ - the action on the geometric fiber of the generic point is pretty easy to understand: We only have to see that adding all roots of $X^3-2$ instead of only one still gives an étale cover which has the advantage of being galois with automorphism group $S_3$. This group acts on the geometric fiber of the generic point like the galois group of $\mathbb{Q}(\sqrt[3]{2}, \zeta_3)$, so it acts on our "small" étale extension above like $S_3$ usually acts on a set with three points. The action of the whole étale fundamental group factors over the action of this smaller group, and we see very concretely that it can't be well defined on the set-theoretic fiber.
What you can try and do to rectify the situation is to define a "smaller" version of the étale fundamental group as the automorphism group of the set-theoretic fiber functor. This would be a subgroup of the étale fundamental group, but it will depend on your choice of basepoint and it won't be naively functorial, but it might work if you decide to work in some category of pointed schemes. What would you need the action for?
