Every elementary abelian p-group occurs as the Frattini quotient of a p-group in which every maximal subgroup is characteristic

Statement
Suppose $$p$$ is a prime number and $$V$$ is an elementary Abelian $$p$$-group (equivalently, $$V$$ is a vector space over the prime field $$\mathbb{F}_p$$. Then, there exists a finite $$p$$-group $$P$$ such that $$P/\Phi(P) \cong V$$ (where $$\Phi(P)$$ denotes the Frattini subgroup of $$P$$) and such that every maximal subgroup of $$P$$ is a characteristic subgroup.

Facts used

 * 1) Bryant-Kovacs theorem

Proof
For the case that $$V$$ is one-dimensional, we can take $$P = V$$. For the case that $$\operatorname{dim}(V) > 1$$, apply fact (1) with the subgroup $$G$$ as the trivial group.