In the book I'm study is written:
A normal subgroup of a group need not be characteristic.
And as an exercise I'm supposed to find an example, it also said that is pretty hard to find one. After trying for two days I wasn't able to find one example.
So, I'm asking for an example of a group $G$ with a normal subgroup $H$ that is not a characteristic of $G$.
I will add some context because maybe it will clarify why the book said it is difficult to find an example: the main problem with the exercise is that until it the book had only covered Group Definition, Subgroups, Lagrange's Theorem and Homomorphisms. So I'm supposed to find a example with such lack of advanced tools.
Best Answer
Consider the additive group $\mathbb{Q}$ of rational numbers. The map $\varphi\colon\mathbb{Q}\to\mathbb{Q}$ defined by $\varphi(x)=x/2$ is readily seen to be an automorphism.
The subgroup $\mathbb{Z}$ is not sent into itself by $\varphi$, because $\varphi(1)=1/2\notin\mathbb{Z}$.
Note that $\mathbb{Q}$ is abelian, so every subgroup is normal.