Wild ramification of nilpotent coverings and coverings of bounded degree

A finite \'etale map between irreducible, normal varieties is called tame, if it is tamely ramified with respect to all partial compactifications whose boundary is the support of a strict normal crossings divisor. We prove that if the Galois group of a Galois covering contains a normal nilpotent subgroup of index bounded by a constant N, then the covering is tame if and only if it is tamely ramified with respect to a single distinguished partial compactification only depending on N. The main tools used in the proof are Temkin's local purely inseparable uniformization and a Lefschetz type theorem due to Drinfeld.