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 or $k=K$.

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....

Perhaps I should also add that probably I can prove some very partial positive results, for some very small and "abelian" subcategories of Schemes. That's the motivation for the question.