mod \(p\) descent for Hilbert modular forms (Q1588360)

Let \(F\) be a totally real number field, and \(k\) a finite field of characteristic \(p>2\). Let \(\rho_F: \text{Gal} (\overline{F}/F)\to \text{GL}_2(k)\) be an absolutely irreducible, continuous and odd representation. Assume that \(\rho_F\) is modular (i.e. arises from a holomorphic Hilbert modular form of some weight and level) and extends to a representation \(\rho_{\mathbb{Q}}: \text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_2(k')\) with \((k':k)< \infty\). Question (Khare). Is the extension \(\rho_{\mathbb{Q}}\) modular? Any such extension is odd and absolutely irreducible, hence, Serre's conjecture implies the answer to this question is ``yes''. The author provides an answer to this question under some assumptions (including the solvability of \(F/\mathbb{Q}\)).