Today in my research, I had to use fairly explicitly the rather tautological property of finite cyclic groups that every normal subgroup is characteristic, i.e. fixed by all automorphisms. This got me wondering:

> do (finite) groups with the property that every normal subgroup is characteristic
> have a name and/or can they be completely
> classified? Generally, has this
> property been investigated at all?

Apart from cyclic groups, some groups possessing the above property that immediately come to mind are simple groups, symmetric groups, [semi-dihedral groups][1], and dihedral groups of twice odd order (however not of twice even order).

This is as far as I got on my short walk home (apart from some false claims, see comments). I suspect that this property might be well studied.

**Edit**:
The reference that Beren Sanders provided in his answer and the references to and from it all deal with $p$-groups. I still haven't been able to find anything about arbitrary finite groups. Some of the questions that $p$-group theorists ask are just not terribly interesting in the case of arbitrary groups. E.g. the paper that Beren Sanders mentioned proves that every finite $p$-group is contained in another finite $p$-group in which every normal subgroup is characteristic. The same statement for arbitrary finite groups is trivial: just embed your group into a symmetric group. I would still be surprised if nobody had tried to say something reasonably general about finite groups with this property.

  [1]: http://en.wikipedia.org/wiki/Quasidihedral_group