Let $L/\mathbb{Q}$ be a dihedral extension of degree $2^{k+1}$, namely that $L$ is Galois over $\mathbb{Q}$ and $\text{Gal}(L/\mathbb{Q}) \cong D_{2^k}$, the dihedral group on $2^{k+1}$ elements.

There is a unique quadratic subfield $K$ of $L$ such that $L/K$ is a cyclic degree $2^k$ extension when $k \geq 2$, corresponding to the unique index-2 cyclic subgroup.

My questions are:

1) For a fixed real quadratic field $K$, how do we parametrize all such $L$ satisfying the above properties, for fixed $k$?

2) How do we determine when $L/K$ is an unramified extension of number fields?