Classification conjecture for Frobenius automorphisms of extended structures on cyclic group algebras

Let $C_n$ be a cyclic group with generator $g$, and let an extended structure on $\Bbbk C_n$ have Frobenius automorphism $\phi$. Let $\omega_n\in\Bbbk$ be any $n$-th root of unity.

Classification conjecture. The only possibilities for $\phi$ are

$$\phi(g)=\pm g\quad\text{or}\quad \phi(g)=\omega_n g^{-1}\qquad\text{when }n\text{ is even},$$

and

$$\phi(g)=g\quad\text{or}\quad \phi(g)=\omega_n g^{-1}\qquad\text{when }n\text{ is odd}.$$

This conjecture proposes a complete classification of the possible Frobenius automorphisms for extended structures on cyclic group algebras, extending the preceding classifications for $C_2$, $C_3$, and $C_4$. Its general validity for all $n$ remains open in the supplied source.