The first level of ${\mathbb{Z}}_p$-extensions and compatibility of heuristics

Let $K$ be an imaginary quadratic field in which the odd prime $p$ does not split. When the $p$-part of the class group of $K$ is cyclic, we describe the possible structures for the $p$-part of the class group of the first level of the cyclotomic ${\mathbb{Z}}_p$-extension of $K$. This allows us to show the compatibility of the heuristics of Cohen–Lenstra–Martinet for class groups with the heuristics of Ellenberg–Jain–Venkatesh for how often the cyclotomic Iwasawa invariant $\lambda$ equals 1.