Normal not implies characteristic

Statement

A normal subgroup of a group need not be a characteristic subgroup.

Example

Let $G$ be any nontrivial group. Then consider $K=G\times G$, viz the external direct product of $G$ with itself. The subgroup $G\times \{e\}$ is a normal subgroup of $K$ (being one of the direct factors).

However, $G\times \{e\}$ is not a characteristic subgroup, because it is not invariant under the automorphism $(x,y)\mapsto (y,x)$ (called the exchange automorphism).

Note that this example also shows that direct factor does not imply characteristic subgroup.