As I said previously, I have some problems in the theory of valuations and places.

Let L/K be a finite (say) Galois extension, F a place of L, and v a valuation of L.
I denote by l and k the residue field of F and of the restriction of F to K resp.
It is known that the map from the decomposition group D of F to the group of automorphisms of the residual field extension l/k is onto.
This means that for every k-automorphism t of l/k, there exists an automorphism s of Gal(L/K) such that tF = Fs.

Also the inertia group of F is exactly the set of automorphisms s of Gal(L/K) such that Fs = F.
But what is known about the set of automorphisms s of Gal(L/K) such that vs = v ? and is there a theorem according to which the map from the decomposition group D of v to the group of automorphisms t of the Abelian ordered group of values of v that fixes the group of values of the restriction of v to K, is onto ? (it is clear that an automorphism s of D induces such an automorphism via the formula tv(x) = v(sx), but is this homomorphism surjective) ?
I have found not hint in Fried and jarden, nor in Bourbaki.