Arithmetic descent of specializations of Galois covers

Given a $G$-Galois branched cover of the projective line over a number field $K$, we study whether there exists a closed point of $\mathbb{P}^1_K$ with a connected fiber such that the $G$-Galois field extension induced by specialization "arithmetically descends" to $\mathbb{Q}$ (i.e., there exists a $G$-Galois field extension of $\mathbb{Q}$ whose compositum with the residue field of the point is equal to the specialization). We prove that the answer is frequently positive (whenever $G$ is regularly realizable over $\mathbb{Q}$) if one first allows a base change to a finite extension of $K$. If one does not allow base change, we prove that the answer is positive when $G$ is cyclic. Furthermore, we provide an explicit example of a Galois branched cover of $\mathbb{P}^1_K$ with no $K$-rational points of arithmetic descent.Comment: We have generalized Theorem 4.1 (in the current numbering) to apply to all cyclic groups. The proof is somewhat dissimilar to the proof of the corresponding statement in the previous version (Theorem 5.1 in v2). Subsequently, some of the numbering has change