On the Class Group of an Imaginary Cyclic Field of Conductor $8p$ and $2$-power Degree

Let $p=2^{e+1}q+1$ be an odd prime number with $2 \nmid q$. Let $K$ be the imaginary cyclic field of conductor $p$ and degree $2^{e+1}$. We denote by $\mathcal{F}$ the imaginary quadratic subextension of the imaginary $(2,\,2)$-extension $K(\sqrt{2})/K^+$ with $\mathcal{F} \neq K$. We determine the Galois module structure of the $2$-part of the class group of $\mathcal{F}$.