# Is the norm element characteristic in modular group rings?

Let $$G$$ be a finite $$p$$- group and let $$\varphi$$ be an automorphism of $$\mathbb{F}_pG$$ as $$\mathbb{F}_p$$-algebras and let $$n = \sum_{g\in G} g$$ be the norm element. Does it follow that $$\varphi(n)=n$$?

Since the $$\mathbb{F}_p$$-vector space spanned by $$n$$ is exactly the annihilator of the maximal ideal, it follows that this vector space is characteristic and thus $$\varphi(n)=\lambda n$$ for some $$\lambda \in \mathbb{F}_p^*$$ and thus we have a group homomorphism: $$Aut(\mathbb{F}_pG)\rightarrow \mathbb{F}_p^*$$. So the question is whether this map is always trivial.