Let $D_n$ be the dihedral group of order $2n$. Then all the quotients of $D_n$ are dihedral as well, and of the form $D_k$ with $k \mid n$. So for a field $K/\mathbb{Q}$ with $\operatorname{Gal}(K/\mathbb{Q}) \cong D_n$, there exists, for any $k \mid n$, a subfield $F \subseteq K$ with $\operatorname{Gal}(F/\mathbb{Q}) \cong D_k$.

My question is about the reverse question. Given a number field $F/\mathbb{Q}$ with $\operatorname{Gal}(F/\mathbb{Q}) \cong D_k$, is there a field $K \supset F$ such that $\operatorname{Gal}(K/\mathbb{Q}) \cong D_n$ for any $n$ a multiple of $k$?

I've been told that this is called the "Galois embedding problem" and is not true for many types of groups. I was wondering if anyone could point me in the right direction for what is known about this in the dihedral case.

Thanks, MC