Let $K = \mathbb{Q}(\sqrt{-D})$ be an imaginary quadratic number field for some $D >0$. Let $H$ be a finite Galois extension of $K$. There is the short exact sequence $$ \operatorname{Gal}(H/K)\to \operatorname{Gal}(H/\mathbb{Q})\to \operatorname{Gal}(K/\mathbb{Q}) $$ Let $\tau\in\operatorname{Gal}(K/\mathbb{Q})$ denote the only non-trivial element of complex conjugation. I want to say that the section $s: \operatorname{Gal}(K/\mathbb{Q})\to \operatorname{Gal}(H/\mathbb{Q})$ by sending $\tau$ to $\tilde{\tau}$ which is also complex conjugation splits the short exact sequence. My reasoning is that this is because $\operatorname{Gal}(H/K)$ does not contain complex conjugation, since it has to fix $K$, which is not totally real.
Now, by assuming this, in what I'm investigating, I've come to a conclusion that doesn't really make sense. I want to check whether or not this assumption is the problem.