It is obvious that if the Frattini quotient of a finite group $G$ is abelian, then $G$ is abelian by nilpotent and that finite nilpotent groups have abelian Frattini quotient. 


I wonder if there is an example of a non-nilpotent finite group with abelian Frattini quotient ? 


A better thing would be to have a nice characterization of such finite groups. 



Are finite nilpotent groups the only finite groups with abelian Frattini quotient?
This is well known, but here is a quick proof anyway.


Suppose that $G/\Phi(G)$ is nilpotent, and let $P \in {\rm Syl}\_p(G)$. Then $P\Phi(G) \unlhd G$ so, by the Frattini Argument, $G = N\_G(P)P\Phi(G)= N\_G(P)\Phi(G)$. Hence $G = N\_G(P)$ by the non-generator property of $\Phi(G)$. So all Sylow subgroups are normal, and $G$ is nilpotent.


