Let be $G, H$ groups and $\Phi(G), \Phi(G)$ their Frattini groups. I'm looking for a conterexample that shows that in general $\Phi(G \times H)= \Phi(G) \times \Phi(H)$ doesn't hold, therefore that there exist $G, H$ such that $\Phi(G) \times \Phi(H)\not \subset \Phi(G \times H)$ (inclusion in other direction is indeed true).