  We improve the existing lower bounds on the order of counterexamples to a
conjecture by P. Schmid, determine some properties of the possible
counterexamples of minimum order for each prime, and the isomorphism type of
the center of the Frattini subgroup for the counterexamples of order 256. We
also show that nonabelian metacyclic p-groups, nonabelian groups of maximal
nilpotency class and 2-groups of coclass two satisfy the conjecture.
