A small disclaimer: My knowledge of algebraic geometry is relatively basic so I will not discuss anything related to scheme directly. But it seems that most of what you are asking has very little to do with algebraic geometry so I will answer your question from a purely topos theoretic perspective.
Moreover, when I say geometric morphism, I talking about the inverse image functor (something is preserved by geometric morphism if it is preserved by the inverse image functor of the geometric morphism)
First your definition of $Aut(P)$ does not make sense, because "aut" is not a functor, if you have a morphism from $P'$ and $P$ and an automorphism of $P$ or $P'$ you have no way to transport it into an automorphism of the other.
Here is how one can define the sheaf or presheaf of automorphism of an object:
Definition of Aut(X), and (non) compatibility to geometric morphism :
First any topos $T$ (the category of sheaves over some site) is cartesian closed, i.e. if $X$ and $Y$ are object of $T$ there is an object $Hom(X,Y)$ characterized by the fact for all object $V$ the morphism from $V$ to $Hom(X,Y)$ are exactly the morphism from $V \times X$ to $Y$.
Note that if you are working within a topos of sheaves, this gives an explicit description of $Hom(X,Y)$ as a sheaf: for any object $c$ of the site $Hom(X,Y)(c)$ is the set of morphism from $X \times \tilde{c}$ to $Y$ where $\tilde{c}$ is the representable sheaf associated to $c$ (the sheafification of the presheaf represented by $c$).
In particular, one gets an object $End(X)$ with the universal property that morphism from $V$ to $End(X)$ is the same as a morphism from $V \times X$ to $X$. One can then define the object $Aut(X)$ which will be the subobject of $End(X)$ of invertible endomorphism, this is done using internal logic and after a small computation on obtain that $Aut(X)$ can be described as follow: a morphism from $V$ to $Aut(X)$ is the same as an invertible morphism from $V \times X$ to $V \times X$ which is compatible to the projection to $V$ (i.e. $p \circ f = p)$, here the functoriality is given as follow, if $W \rightarrow V$ is a morphism then $X \times W$ is the pullback of $X \times V$ along the map $W \rightarrow V$, hence any automorphism of $X \times V$ over $V$ can be pulledback into an automorphism of $X \times W$ over $W$.
Now it is a general fact that this construction of $Aut(X)$ is in general not compatible with pullback along geometric morphism or sheafification functor. Hence in general it will not be the case $Aut(a(P)) = a(Aut(P))$ it might be the case for some very special site, but I see no reason for it to be true for the site you are using.
Very informally, the reason this is not the case is because sheafification and geometric morphism only preserve finite product and not infinite product. And $End(X)$ is morally $X^X$ so you unless you have some finiteness condition on $X$ (typically cardinal finite), $End(X)$ will not be preserved by geometric morphism. In your second question you are concerned with $G$ automorphism of an object which is a "finitely generated" $G$ object and this is why it is going to work.
The second case discussed in the comment:
Sill in a general topos, if $G$ is a group object and $H$ is a subgroup, you can always define a quotient $G/H$ which is a $G$-object with the expected universal property: if $X$ is any other $G$-object a morphism of $G$-object from $G/H$ to $X$ is the same as a morphism from $G$ to $X$ which is constant on $H \subset G$.
The existence of $G/H$ can be deduced either from internal logic or just from the co-completness of every topos, but this kind of universal property does not produce a description of $G/H$ as a sheaf anymore (because we don't know what are morphisms from an object $V$ to $G/H$, only morphism out of $G/H$). But one can prove that the construction of $G/H$ is compatible to sheafification and more general geometric morphism, meaning that $f^*(G/H)=f^*(G)/f^*(H)$ which implies that:
-the sheaf $G/H$ is the sheafification of the presehaf $G/H$.
-the presheaf $G/H$ is the object wise quotient $G(c)/H(c)$ because $X \mapsto X(c)$ is the inverse image functor of a geometric morphism $Set \rightarrow prsh(C)$.
Still in the same situtation $H \subset G$ one can also construct a "normalizer" $N_G(H)$ (also using internal logic) which is a subobject of $G$ satisfying the universal property:
A morphism from $V$ to $N_G(H)$ is the same as a morphism $f : V \rightarrow G$ such that the map from $H \times V$ to $G$ defined by $(h,v) \mapsto f(v) h f(v)^{-1}$ factor into $H$.
i.e. it is just the internal interpretation of $\{ g \in G | \forall h \in H, g h g^{-1} \in H \}$.
This mean that, as for $Aut(X)$ and $End(X)$ above you have a description of $N_G(H)$ as a sheaf (and if $G$ and $H$ are sheave then $N_G(H)$ is indeed a sheaf).
one easily check that $N_G(H)$ is a group object (a subgroup of $G$ ) containing $H$.
Now in any topos, you have the following results:
$$ Aut_G(G/H) \simeq N_G(H)/H $$
where $Aut_G(X)$ denotes the sub-object of $Aut(X)$ of morphism compatible to the $G$ action, i.e. a morphism from $V$ to $Aut_G(X)$ is an automorphism of $X \times V$ compatible to the projection on the second component and to the $G$ action on the first component.
The isomorphism is implemented by the natural action of $N_G(H)$ on $G/H$ induced by the action of $G$ and $G$. The proof of the isomorphism is very easy when one knows internal logic: the usual set theoretic proof of this isomorphism is fully constructive hence it is true in any topos.
This result is true both in the presheaf topos and in the sheaf topos, but the right hand side is known to be compatible with sheafification and geometric morphism (in the sense that $N_G(H)/H$ in the sheaf topos is the sheafification of $N_G(H)/H$ computed in the presheaf topos).
This answer your question in the comment.
The $aut_G(X)$ you are talking about is $Aut_G(G/H)$ in the sheaf topos and $a(N_G(H)/H )$ is the sheafification of $N_G(H)/H$ in the presheaf topos and hence it is $N_G(H)/H$ in the sheaf topos, and the isomorphism you are aksing is exactly the one I mentioned above.
