Grothendieck calls a "discretification" of a profinite group $\hat G$, a
discrete group $G$ whose profinite completion is isomorphic to $\hat G$.

Does Grothendieck also  define a notion of the "discretification"
of a functor to profinite groups, and particulary that of the
algebraic fundamental group functor?

The automorphisms of the geometric point act on the discretifications
of the fundamental group functor; and thus one can ask: 

>are all discretifications of the the algebraic fundamental group *functor*
related by automorphisms of the geometric point? To simplify the question,
one may abelianise the functor and/or restrict it to a full subcategory of
schemes $X\longrightarrow S$, $S=Spec k$ a number field.

Here is the same question with some more notation and detail.
Please excuse my poor notation: I am not very familiar with the language of schemes. 

Say a functor F to finitely generated Abelian groups is a _discretification_
of a functor F' to profinite groups iff for every $X$, $F(X)$ is 
a discretification of $F'(X)$. Equivalently, one is given an equivalence $\hat F==>F'$
(where $\hat F(X)=\hat{F(X)}$ is the profinite completion of $F(X)$.)

The algebraic fundamental group is, or at least defines,
a functor $pi_1^{alg}:$ Spec K\Schemes $\longrightarrow ProfiniteGroups$
from the category Spec K\Schemes $= Mor (Spec K, Schemes)$
of Schemes cosliced over a geometric point $Spec K$.

There is a canonical Aut(K/Q) action on Spec K\Schemes
which extends to an action on the discretifications
of this functor. Did Grothendicek conjecture
anything about this action?

Is there a better way to state this question? E.g.
using the fact that for K the field of algebraic numbers
$Aut(K/Q)=pi_1^{alg}(Spec Q, SpecK)$.

One can modify the question by replacing the category
of  Schemes by a subcategory of Schemes over Spec k
or some other scheme, and then considering double
cosets of Aut(K/Q) and Aut(k/Q) action; as an example,
one may consider the full subcategory of etale covers 
of direct products of moduli spaces of curves....