The automorphism group of a valued field of generalised formal power series

Let $ k be a field, G totally ordered abelian group and \mathbb K = k((G)) the maximal field of generalised power series, endowed with canonical valuation v $. We study \mathrm{-Aut} preserving automorphisms subfield k(G)\subseteq K\subseteq $, where k(G) is fraction ring k[G] Under assumption that satisfies two lifting properties we are able to generalise refine Hofberger's decomposition \mathrm{-Aut}\mathbb prove structure theorem decomposing v\mathrm{-Aut} into 4-factor semi-direct product notable subgroups. then identify large class Hahn fields satisfying aforementioned properties. Next focus on strongly additive give an explicit description internal in terms groups homomorphisms \mathrm{Hom}(G,k^\times) k^\times \mathrm{Hom}(G,1+I_K) 1-units Finally, specialise our results some relevant special cases. In particular, extend work Schilling Laurent series Deschamps Puiseux series.