Minimal fields of definition for Galois action

Let K   be a field, and let f:X→Yf:X→Y be a finite étale cover between reduced and geometrically irreducible K  -schemes of finite type such that fKsfKs is Galois. Assuming f admits a Galois K  -form f¯:X¯→Y, we use it to analyze fields of definition over K for the Galois property of f and the presence of K-points in general K  -forms f′:X′→Yf′:X′→Y over Y(K)Y(K).Additionally, we show that if K is Hilbertian, the group G   is non-abelian, and the base variety is rational, then there are finite separable extensions L/KL/K such that some L  -form of fLfL does not descend to a cover of Y.