All groups in this paper are finite. Given a group G with subgroups \(H_ 0\), H and \(G_ 0\) such that \(H_ 0\trianglelefteq H\), we call \(G_ 0\) a relative normal complement of H over \(H_ 0\) in case \(G_ 0\trianglelefteq G\), \(G=G_ 0H\) and \(H_ 0=G_ 0\cap H\). Especially when \(H_ 0=1\) and H is a \(\pi\)-Hall subgroup in G, then \(G_ 0\) is called normal \(\pi\)-complement of H in G. In this paper, the author gives a sufficient condition for the existence of a normal complement of H over \(H_ 0\) in G and necessary and sufficient conditions for the existence of a normal \(\pi\)-complement of H in G.