On the Frattini subgroup of a finite group

We study the class of finite groups $G$ satisfying $\mathrm{\Phi }(G\mathrm{/}N)=\mathrm{\Phi }(G)N\mathrm{/}N$ 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.