On the Frattini subgroup of a finite group

We study the class of finite groups G   satisfying Φ(G/N)=Φ(G)N/NΦ(G/N)=Φ(G)N/N for all normal subgroups N of G. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.