  Let $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension field of an
algebraic number field $k$. Moreover, we take a $\mathbb{Z}_p$-extension
$K_\infty$ over $k_\infty$. In this paper, we study the behavior of the
$p$-part of the class number of certain intermediate fields of $K_\infty /k$.
