Suppose $G$ is a finite group, such that $\Phi(G)$ is non-abelian. Does there always exist such prime $p$, that $p^5 | |G|$? Here $\Phi$ stands for Frattini subgroup.
Using the same method, as the one used in the answer to “If $|G|=p^3q^2$ then $\Phi(G)$ is cyclic for primes $p\neq q$.”, we can reduce this question to the following ones:
Suppose $G$ is a finite group, such that $\Phi(G) \cong D_4$. Is it always true, that $32 | |G|$?
Suppose $G$ is a finite group, such that $\Phi(G) \cong Q_8$. Is it always true, that $32 | |G|$?
Suppose $G$ is a finite group, such that $\Phi(G) \cong (C_p \times C_p) \rtimes C_p$ for some odd prime $p$. Is it always true, that $p^5 | |G|$?
Suppose $G$ is a finite group, such that $\Phi(G) \cong C_{p^2} \rtimes C_p$ for some odd prime $p$. Is it always true, that $p^5 | |G|$?
The answer to the main question is positive iff the answer is positive in all those four particular cases.
A problem, similar to those “reduced ones” was solved for $C_p \times C_p$ here: A question about Frattini subgroup of specific form However, the solution seems to rely on the structure of $Aut(C_p \times C_p)$ and thus the method does not seem to be directly applicable in our case (though, probably, something similar may be...).
And I do not know how to proceed further.