Normal closure of the nonnormal factor of Holomorph of a Cyclic group

Question

Let $C_n$ be the cyclic group of order $n$. Then, we can consider the holomorph $G=C_n\rtimes Aut(C_n)$. let $H$ be such that $Aut(C_n)\leq H\trianglelefteq G$. Is it necessarily the case that $H$ is of index $1$ or $2$ in $G$.

I'm asking because this came up because of this question: Galois extension of real subfield is of degree at most $2$?

Answer 1

Yes. Let $G = C_n \rtimes A$ with $A = {\rm Aut}(C_n)$ and $C_n = \langle g \rangle$. Since there exists $a \in A$ with $a^{-1}ga=g^{-1}$, we have $g^2 \in [A, C_n]$, and $A \le H \unlhd G$ implies $[A,C_n] \le H$, so $\langle g^2, A \rangle \le H$ and $|G:\langle g^2, A \rangle| \le 2$.