I define the notion of "Galois class of L functions" in the following way: $A$ is a Galois class of L functions if and only if the follwing three conditions hold simultaneously:

1) every element of $A$ belongs to the Selberg class and factors in a unique fashion in a product of primitive functions of this class

2) the constant map equal to $1$ belongs to $A$

3) whenever $F$ and $G$ are in $A$, then so is $F.G$.

Then I define the group $Aut(A)$ of the automorphisms of a Galois class of L functions $A$ as the group of bijections $\phi$ from $A$ to itself such that the following conditions simultaneously hold:

1) for all $F$ in $A$, $d_{\phi(F)}=d_{F}$ where $d_{F}$ is the degree of $F$

2) $\phi$ maps any primitive function of $A$ to a primitive function of $A$

3) for all $F$, $G$ in $A$, $\phi(F.G)=\phi(F)\phi(G)$.

Now let $M$ be the maximal Galois class of L functions, $F$ a primitive element of $M$. I denote by $<F>$ the Galois class of L functions generated by $F$, of the form $\{F^{n},n\in\mathbb{N}\}$.
I also define the "structure group" of $<F>$, denoted as $Str( <F> )$, as the group of automorphisms of $M$ preserving $F$. Is it true that there exists $\ell\in\mathbb{P}\cup\{\infty\}$ such that $Str( <F> )$ is isomorphic to (a subgroup of) the absolute Galois group of $\mathbb{Q}_{\ell}$?

Thanks in advance.

EDIT: the desired subgroup of $Gal(\overline{\mathbb{Q}_{\ell}}/\mathbb{Q}_{\ell})$ might be of index $1$ or $2$ depending on whether $F$ is self-dual or not.