$2$-parts of class groups in thin families

We will give an overview of what is known about the statistics of $2$-parts of class groups of quadratic number fields parametrized by discriminants of the form $dp$, where $d$ is fixed and $p$ varies over the set of prime numbers. In these thin families, the map sending a prime $p$ to the $2^k$-rank of the class group of $\mathbb{Q}(\sqrt{dp})$ is Frobenian for $k\leq 3$ (a result of Peter Stevenhagen) and likely not Frobenian for $k\geq 4$. We will summarize a method that gives some statistical results about the $16$-rank in the absence of governing fields. The $32$- and higher $2$-power-ranks remain out of reach despite Alexander Smith's recent breakthroughs.