  Let $K/\mathbb{Q}$ be a quadratic extension. In this paper we study the
$4$-rank of the class group $\text{Cl}(K(\sqrt{n}))$, where $n$ varies over
squarefree rational integers. We show that for $100\%$ of squarefree $n$, the
$4$-rank is given by an explicit formula involving the $2$-rank of
$\text{Cl}(K)$ and the number of prime factors of $n$ which are inert in
$K/\mathbb{Q}$.
