Let $K$ be an imaginary quadratic number field with $p$-Sylow-class group $A(K)$ and $L/K$ be an unramified cyclic extension of $K$ of degree $p$ ($p$ prime). Then I am looking for heuristics on

$ker(N_{L/K}:A(L)\rightarrow A(K))$,

where $N_{L/K}$ is the usual norm map on ideal classes. Numerical data suggest that:

$|KerN_{L/K}|\le p^2$ with high probability. But that is rather unprecise. Are there any more satisfactory known results ?

There is a result by Wittmann who gives some heuristics for cyclic $p$-extensions of $\mathbb{Q}$ and I wonder if one can possibly generalize his ideas to the above scenario.