On some applications of finite galois stable linear groups

We consider a Galois extensions E/F and realization fields of finite abelian subgroups G ⊂ GL _n(E) of a given exponent t. Let us assume that G is stable under the natural operation of the Galois group of E=F. It is proven that under some reasonable restrictions for n any E can be a realization field of G, while if all coefficients of matrices in G are algebraic integers there are only finitely many fields E of realization having a given degree d for prescribed integers n and t or prescribed n and d: Some related results and conjectures are considered.