This question is a follow-up to Galois classes of L-functions. My goal here is to make things clearer.

Definition 1

Let $A$ be a subclass of the Selberg class containing $s\mapsto 1$, closed under products, and such that every element of $A$ can be factored in a unique fashion in a product of primitive elements of $\mathcal{S}$, these primitive elements belonging to $A$. Such a subclass of $\mathcal{S}$ will be called a Galois class of L-functions.

Definition 2

Let $A$ be a Galois class of L-functions. $T$ is an automorphism of $A$ iff the following properties simultaneously hold true:

1) $T$ is a bijective map from A to itself

2) $T$ maps a primitive element of $A$ to a primitive element of $A$

3) for all $F$ in $A$, the degree of $F$ and the degree of $T(F)$ are the same

4) for all $F$, $G$ in $A$, $T(F.G)=T(F).T(G)$

Let $M$ be the maximal Galois class of L functions, $F$ an element of $M$. Let's denote $G_{S}(F)$ the group of complex isometries preserving globally the multiset of non trivial zeros of $F$, $G_{F}$ the group of automorphisms of $M$ preserving $F$.

Hadamard's factorization theorem says that knowing $F$ is equivalent to knowing its multiset of non trivial zeros. Therefore, preserving this multiset is equivalent to preserving $F$.

My question is: does this mean that among the automorphisms of $M$ preserving $F$, there exist "good" automorphisms of $M$ preserving $F$ such that if $G_{F}^{good}$ is the group of such "good" automorphisms, then there exists a group homomorphism $\rho: G_{F}^{good}\to G_{S}(F)$ that is actually an isomorphism?

Thanks in advance.

EDIT September 18th 2013: maybe one could establish the existence of the desired isomorphism asking the question in the framework of category theory, for example showing that the two categories $\mathcal{C}$ and $\mathcal{D}$ consisting of elements of $M$ or their multisets of zeros with the desired morphisms between these objects are equivalent, hence showing that the corresponding groups of automorphisms are isomorphic. In other words, I expect the groups $Aut_{\mathcal{C}}(F)$ and $Aut_{\mathcal{D}}(Z(F))$ for $F$ an object of $\mathcal{C}$ and $Z(F)$ the corresponding object of $\mathcal{D}$ to be, respectively, equal to $G_{F}^{good}$ and $G_{S}(F)$. Is there any hope to prove this rigorously?