Defining the notion of "Galois class of L-functions" as a set of automorphic L-functions belonging to the Selberg class closed under the usual product and the Rankin-Selberg convolution and containing the constant map equal to $1$ and the Riemann zeta function, say $\mathcal{G}(F)$ is the Galois class of L-functions generated by $F$ if it is the smallest Galois class of L-functions containing $F$.

Do(es) the so far proven multiplicity one theorem(s) imply that an element of a Galois class of L-functions is entirely determined by its set of Satake parameters at all but finitely many places? If so, is the group of automorphisms of $\mathcal{G}(F)$ for a given $F$ (hence bijective maps from $\mathcal{G}(F)$ to itself preserving the usual product and RS convolution and sending $s\mapsto 1$ to itself and $\zeta$ to itself) naturally isomorphic to the Galois group $Gal(K_{F}/Q)$ where $K_{F}$ is the compositum of the number fields $K_{p}(F)$ obtained by adjoining to $Q$ the Satake parameters of $F$ at $p$ for all but finitely many $p$?

Would this imply that the group $Aut(\mathcal{M})$ where $\mathcal{M}$ is the maximal Galois class of L-functions is naturally isomorphic to the absolute Galois group of the rationals?