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