Let $G$ be a finite supersolvable group with trivial Frattini subgroup. Is it true that all Sylow subgroups of $G$ are elementary abelian?

**EDIT**: Many Thanks to Derek for his answer. Let me say some words on the motivation.
I am asked of:
Is it true that in a finite supersolvable group with trivial Frattini subgroup, every subgroup is supplemented?
A subgroup $H$ of a group $G$ is called supplemented if there exists a proper subgroup $K$ of $G$ such that $G=KH$. If $K\cap H=1$, $H$ is called complemented.
I learnt that in a finite group if every subgroup is supplemented then every subgroup is complemented [This is Corollary 3.7 of L.-C. Kappe and J. KIRTLAND, SUPPLEMENTATION IN GROUPS, Glasgow Math. J. 42 (2000) 37-50] and it follows from what is mentioned in the paragraph after the statement of Corollary 3.7 in the latter paper that in a supersolavble group, every subgroup is supplemented if all Sylow subgroups of the group are elementary abelian. This is a result due to P. Hall, Complemented groups, J. London Math. Soc. 12 (1937), 201-204.