Assuming Selberg's orthonormality conjecture and following Automorphisms of the Selberg class, I define automorphisms of the Selberg class $\mathcal{S}$ as bijective maps $f$ from $\mathcal{S}$ to itself such that:

1. $f$ maps a primitive function of $\mathcal{S}$ to a primitive function of $\mathcal{S}$,

2. $f$ maps a function of degree $d$ to a function of degree $d$,

3. for every $F$ in $\mathcal{S}$, $n_{f(F)}=n_F$, where $n_F $is the integer involved in Selberg's conjecture A,

4. if $F=F_{1}^{e_1}.F_{2}^{e_2}.F_{k}^{e_k}$, then $f(F)=f(F_1)^{e_1}.f(F_2)^{e_2}....f(F_k)^{e_k}$ (so that $f$ is "completely multiplicative").

The set of all such maps forms a group $Aut(\mathcal{S})$ under composition, the structure of which has been determined by Denis Chaperon de Lauzières in the link given above.

Now let $F$ be an element of $\mathcal{S}$ and let's define $G_{F}$ as the subgroup of $Aut(\mathcal{S})$ whose every element preserves $F$.
Is it true that $F$ is a primitive element of $\mathcal{S}$ of degree $d$ if and only if there exists a vector space $V$ and a group morphism $\rho: G_{F}\to GL(V)$ such that $(V,\rho)$ is an irreducible representation of $G_{F}$ of degree $d$?

EDIT: I add the following question: considering that $G_{F}$ is somewhat analogous to the absolute Galois group of a field $\mathbb{K}$, the set $\{F^{n},n\ge 0\}$ playing the role of this field and $\mathcal{S}$ the role of its separable closure, and assuming Langlands' reciprocity conjecture, is it possible to prove that every primitive element of the Selberg class is an automorphic L-function?

Thanks in advance.