Let $F$ be a number field and $E/F$ a Galois extension. Suppose we have a representation $\rho_E : Gal(\overline{F}/E) \rightarrow GL_n(\overline{Q}_p)$. My question is : what are sufficiant conditions so that $\rho_E$ can be extended to a representation $ Gal(\overline{F}/F) \rightarrow GL_n(\overline{Q}_p)$ ?

A necessary condition is that $\rho_E$ is invariant under $Gal(E/F)$. This paper (last line of page 1)

http://www.institut.math.jussieu.fr/projets/fa/bpFiles/GaloisPatching_Harris.pdf

claims that such an extension exists if moreover $\rho_E$ is irreducible and $E/F$ is cyclic of prime ordre, but I don't know why it is true.