The kernel generating condition and absolute Galois groups

For a list L of finite groups and for a profinite group G, we consider the intersection T(G) of all open normal subgroups N of G with G/N in L. We give a cohomological characterization of the epimorphisms π:S→G of profinite groups (satisfying some additional requirements) such that π[T(S)]=T(G). For p prime, this is used to describe cohomologically the profinite groups G whose nth term G_(n,p) (resp., G^(n,p)) in the p-Zassenhaus filtration (resp., lower p-central filtration) is an intersection of this form. When G=GF is the absolute Galois group of a field F containing a root of unity of order p, we recover as special cases results by Minac, Spira and the author, describing G_(3,p) and G^(3,p) as T(G) for appropriate lists L.