Commuting-liftable subgroups of Galois groups

Let n denote either a positive integer or ∞, let ℓ be a fixed prime and let K be a field of characteristic different from ℓ. In the presence of sufficiently many roots of unity, we show how to recover much of the decomposition/inertia structure of valuations in the [special characters omitted]/ℓn-elementary abelian Galois group of K, while using only the group-theoretical structure of the [special characters omitted]/ℓn-abelian-by-central Galois group of K whenever N is sufficiently large with respect to n. Moreover, if n = 1 then N = 1 suffices, while if n ≠ ∞, we provide an explicit N0 ≠ ∞, as a function of n and ℓ, for which all N ≥ N 0 suffice above. In the process, we give a complete classification of so-called commuting-liftable subgroups of elementary-abelian Galois groups and prove that they always arise from valuations