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. Isr. J. Math. 172, 357–369 (2009). Zbl 1188.20014
As a non-expert I have nothing more to add.
