Let $F$ be a nonarchimedian local field. Since the Weil group $W_F$ is a dense subgroup of $G_F=Gal(\bar{F}/F)$, it's clear that restriction gives an injection $Irr(G_F)\rightarrow Irr(W_F)$ of irreducible (complex) representations.
In some notes of Prasad and Raghuram (found here: http://www.math.tifr.res.in/~dprasad/ictp2.pdf), they assert that the representations are in fact the same "perhaps after a twist." This fact is crucial for their argument that reduces establishing LLC for $GL_n$ to the case of supercuspidal representations. But I'm not sure I understand what it means.
So my question is: how does one take a representation of $W_F$ and twist it in such a way that it can be extended to a representation of $G_F$?