Galois stability, integrality and realization fields for representations of finite Abelian groups

For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups G ⊂ GL_n(E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E = F(G) that are obtained via adjoining all matrix coefficients of all matrices g ε G to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E = F(G) for prescribed integers n and t or prescribed n and d.