Ive been trying to attack this question from different point of view but i cant make it. Basicly I started thinking that Frattini was not normal, i was trying to get a counterexample but all the groups I try failed.
Now I am convinced that The Frattini subgroup is normal but i cant prove it.
I would appreciate any help
The Frattini subgroup is the intersection of all maximal subgroups (if there aren't any, it is the whole group). Now, if $M$ is maximal in $G$ then also any conjugate $M^g$ is maximal in $G$. Can you prove that first and then prove your post?