> The question in which $p$-groups all normal subgroups are characteristic is a fairly old problem in the theory of finite $p$-groups. It seems difficult to assess because of the fact that neither subgroups nor factors of a group which has none but characteristic normal subgroups need retain this property.

These are the first two sentences of the following 2009 paper published in the Israel Journal of Mathematics:

B. Wilkens. [$p$-groups without noncharacteristic normal subgroups.][1] Isr. J. Math. 172, 357–369 (2009). [Zbl 1188.20014][2]

As a non-expert I have nothing more to add.


  [1]: https://doi.org/10.1007/s11856-009-0078-x
  [2]: https://zbmath.org/?q=an:1188.20014