Let $p$ be an odd prime number and let $2^{e+1}$ be the highest power of 2 dividing $p - 1$. For $0 \leq n \leq e$, let $k_{n}$ be the real cyclic field of conductor $p$ and degree $2^{n}$. For a certain imaginary quadratic field $L_{0}$, we put $L_{n} = L_{0} k_{n}$. For $0 \leq n \leq e - 1$, let $\mathcal{F}_{n}$ be the imaginary quadratic subextension of the imaginary $(2, 2)$-extension $L_{n+1}/k_{n}$ with $\mathcal{F}_{n} \neq L_{n}$. We study the Galois module structure of the 2-part of the ideal class group of the imaginary cyclic field $\mathcal{F}_{n}$. This generalizes a classical result of Rédei and Reichardt for the case $n = 0$.