In Bushnell and Henniart, The Local Langlands conjecture for GL(2), there is a proposition on p. 184 in which they prove the following:

Let $F$ be a non-archimedean local field, $\mathcal W_F$ its Weil group, and $\tau$ an irreducible smooth representation of $\mathcal W_F$. Then, if $\tau(\mathcal W_F)$ is finite, it can be extended to an irreducible smooth representation of $\Omega_F = Gal(\overline F/F)$.

In the proof they fix a Frobenius element $\Phi\in\mathcal W_F$ and write any element $\omega\in \Omega_F$ as $\omega = \Phi^a\sigma$ (for some $a\in \hat{\mathbb Z}$ and $\sigma\in I_F$). My questions is: which element of $\Omega_F$ is meant by $\Phi^a$? Of course, there is an exact sequence $1\to \mathcal I_F\to \Omega_F\to Gal(F^{\text{unr}}/F) = \hat{\mathbb Z}\to 1$ but how is $\Phi$ used to cook up an element of $\Omega_F$ in the fiber of an element $a\in \hat{\mathbb Z}$?

Thanks in advance!

aFrobenius element [of $\mathcal W_F$]" (as opposed to "fixtheFrobenius element [of $\mathcal W_F/\mathcal I_F$]"). Also, are your $\mathcal W_F$ and $\Omega_F$ the same? $\endgroup$ – LSpice Jul 6 '18 at 14:19