Normal subgroups vs characteristic subgroups It is well known that characteristic subgroups of a group $G$ are normal. Is the converse true?

I think the simplest example is the Klein four-group, which you can think of as a direct sum of two cyclic groups of order two, say $A\oplus B$.  Because it is abelian, all of its subgroups are normal.  However, there is an automorphism which interchanges the two direct summands $A$ and $B$, which shows that $A$ (and $B$) are normal, but not characteristic.  (In fact, the other non-trivial proper subgroup, generated by the product of the generators of $A$ and $B$ also works.)
