Automorphisms of vertically simple complex multisets

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Hello,

I defined automorphisms of the Selberg class in http://mathoverflow.net/questions/50581/automorphisms-of-the-selberg-class
Let's call "structural group of the extension $L$ of $K$", denoted by $Str(L/K)$, the group of automorphisms of $L$ which coincide with identity on $K$, whenever $K$ is a substructure of the structure $L$: for example $K$ may be a subring of a ring $L$, or a submonoid, or whatever you want. When $K$ is a subfield of a field $L$, one gets $Str(L/K)=Gal(L/K)$. Let $S$ denote the Selberg class, and $S'$ the set of self-dual functions in $S$. Following Andrew D.Droll's recent thesis (http://qspace.library.queensu.ca/jspui/bitstream/1974/7352/1/Droll_Andrew_D_201207_PhD.pdf), forall $F\in S$, let's denote by $Z(F)$ the "vertically simple complex multiset" of non-trivial zeroes of $F$. Moreover, let $Z_S$ denote `$\bigcup_{F\in S}\{ Z(F)\}$` and `$Z_S'=\bigcup_{F\in S'}\{ Z(F)\}$`.

My question is: is there a natural way to define automorphisms of vertically simple complex multisets such that $Str(S/S')$ and $Str(Z_S/Z_S')$ are canonically isomorphic? If so, are such automorphisms of vertically simple complex multisets necessarily isometries?

Thanks in advance and happy birthday to this website.

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