On finite linear groups stable under Galois operation

We consider a Galois extension E/F of characteristic 0 and realization fields of finite abelian subgroups G ⊂ GL_n(E) of a given exponent t. We 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.