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