  Let N be a normal subgroup of a finite group G. Let N\le H\le G such that N
has a complement in H and (|N|,|G:H|)=1. If N is abelian, a theorem of
Gasch\"utz asserts that N has a complement in G as well. Brandis has asked
whether the commutativity of N can be replaced by some weaker property. We
prove that N has a complement in G whenever all Sylow subgroups of N are
abelian. On the other hand, we construct counterexamples if Z(N)\cap N'\ne 1.
For metabelian groups N, the condition Z(N)\cap N'=1 implies the existence of
complements. Finally, if N is perfect and centerless, then Gasch\"utz' theorem
holds for N if and only if Inn(N) has a complement in Aut(N).
