Seems highly unlikely to me. Here's a possible strategy for a counterexample. Let $K$ be a $p$-adic field and let $L$ be its maximal tamely ramified extension. Then $\Gamma:=Gal(L/K)$ is a well-understood group and it's not abelian. Find two elements that don't commute and finite extensions $M/N/K$ in $L$ such that the elements fix $N$ pointwise, and restrict to commuting elements of $Gal(M/N)$ with the property that no lifts of these elements to $Gal(L/N)$ commute. This is now just a messy problem in group theory. Then $M$ and the automorphisms, if you can find them, give a counterexample.
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Kevin BuzzardMay 27 '12 at 20:08

Why do we even expect the endomorphisms to extend at all to the algebraic closure?
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J. MartelMay 28 '12 at 20:14

In your setting, an appropriate justification for the existence of an extension should use your hypotheses that the field is complete and discrete valued.
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J. MartelMay 28 '12 at 20:16