Let $\widehat{F\_2}$ be the free profinite group of rank 2. Is its Frattini subgroup $\Phi(\widehat{F\_2})$ trivial?


I know that the Frattini subgroup of a pro-$p$ group is open. On the other hand, the Frattini subgroup of the abstract free group $F\_2$ is trivial.


The general profinite case $\widehat{F\_2}$ however seems to resist my googling attempts.


I believe $\Phi(\widehat{F\_2})$ being trivial is equivalent to - Every finite 2-generated group $G$ is the quotient of a finite 2-generated group $G'$ with $\Phi(G') = 1$.


If $\Phi(\widehat{F\_2})$ isn't trivial, is there a description of the Frattini quotient $\widehat{F\_2}/\Phi(\widehat{F\_2})$ ?



Is the Frattini subgroup of a free profinite group trivial?
It is trivial. This is a special case of Corollary 8.7.5 of the book of Ribes and Zalesskii. It uses that every proper open normal subgroup of a closed normal subgroup of a free profinite group is free profinite and that the Frattini subgroup is pronilpotent. 


