I tend to ask questions on mathstackexchange, because I don't feel adequate yet for mathoverflow. I had previously asked this question there (I have now deleted it), where it was quite popular, but it didn't seem like anybody knew the answer. I thought that perhaps this question should graduate to mathoverflow. Here it is, as it had originally appeared:

I've been playing around with the $p$-adics, and I wondered about the structure of their Galois group.

We have the short exact sequence:

$$1\rightarrow Gal(\bar{\mathbb{Q}}_p/\mathbb{Q}_p^{un})\rightarrow Gal(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)\rightarrow Gal(\mathbb{Q}_p^{un}/\mathbb{Q}_p)\rightarrow 1$$

I as wondering if this exact sequence is split. I.e., is it true that you can lift the Frobenius automorphism of $Gal(\mathbb{Q}_p^{un}/\mathbb{Q}_p)$ to $Gal(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$? How would the lifting of this automorphism act on individual elements of $\bar{\mathbb{Q}}_p$?