Class number behavior in a two-tiered tower of ${\mathbb{Z}}_p$-extensions

Let $k_{\mathrm{\infty }}$ be the cyclotomic ${\mathbb{Z}}_p$-extension field of an algebraic number field $k$. Moreover, we take a ${\mathbb{Z}}_p$-extension $K_{\mathrm{\infty }}$ over $k_{\mathrm{\infty }}$. In this paper, we study the behavior of the $p$-part of the class number of certain intermediate fields of $K_{\mathrm{\infty }}\mathrm{/}k$. We also consider the structure of the unramified Iwasawa module of $K_{\mathrm{\infty }}\mathrm{/}{k}_{\mathrm{\infty }}$ for several cases.