Automorphisms of group ring

Question

Let $C_n$ denote the cyclic group of order $n$ and let $R=\mathbb{Z}[C_n]$ be its group ring. Any automorphism on $C_n$ defines a ring automorphism of $R$.

Are these all ring automorphisms of $R$?

Equivalently asked, for a given ring automorphism $\phi:R\to R$ is it true that $\phi(C_n)\subset C_n$?

Answer 1

No, let $n = 2$ and let $C_2 = \langle g \rangle$. Then consider $\Phi: R \to R, a + bg \mapsto a - bg$. This is a ring automorphism but it is not induced by any automorphism of $C_2$.