Regular representation of the Galois groups

Question

Let $G_\overline{\mathbb Q}$ the absolute Galois group of $\mathbb Q$ and let $G'$ the subgroup of order $2$ generated by complex conjugation.

Now, I've show that $G_\overline{\mathbb Q}/G'$ is a space Hausdorff, compact and totally disconnected. Also the space $G_\overline{\mathbb Q}/G'$ has a regular measure that comes from of the haar measure of $G_\overline{\mathbb Q}$ because $G'$ is discrete.

Consider the regular representation $$\varphi_1:G_\overline{\mathbb Q}\to L^2(G_\overline{\mathbb Q}),$$ and the induced representation by multiplication $(\sigma f)(z)=f(\sigma^{-1}z)$ $$\varphi_2:G_\overline{\mathbb Q}\to L^2(G_\overline{\mathbb Q}/G').$$

My question is if the representations $\varphi_1$ and $\varphi_2$ are isomorphics? or what is the diference between this two representations.

Thanks you all.

Answer 1

No, $\varphi_1$ and $\varphi_2$ are not isomorphic: $\ker(\varphi_1)=\{1\}$ while $\ker(\varphi_2)=G'$.