Let $G$ be a finite $p$-group. Suppose that $G$ has only one characteristic subgroup(nontrivial proper). What possible structures does $G$ have? 



Characteristic subgroups of $p$-groups
Have a look at :


S. P. Glasby, P. P. Pálfy, Csaba Schneider, p-Groups with a unique proper non-trivial characteristic subgroup, Journal of Algebra, Volume 348,Pages 85-109


<http://www.sciencedirect.com/science/article/pii/S0021869311005552>


