Gerth's heuristics for a family of quadratic extensions of certain Galois number fields

Gerth generalised Cohen-Lenstra heuristics to the prime $p=2$. He conjectured that for any positive integer $m$, the limit $$\underset{x\to \mathrm{\infty }}{\lim }\frac{\sum _{\genfrac{}{}{0px}{}{0<D\le X,}{\text{squarefree}}}\mathrm{\mid }{\mathrm{C}\mathrm{l}}_{\text{Q}(\sqrt{D})}^2\mathrm{/}{\mathrm{C}\mathrm{l}}_{\text{Q}(\sqrt{D})}^4{\mathrm{\mid }}^m}{\sum _{\genfrac{}{}{0px}{}{0<D\le X,}{\text{squarefree}}}1}$$ exists and proposed a value for the limit. Gerth’s conjecture was proved by Fouvry and Kluners in 2007. In this paper, we generalize their result by obtaining lower bounds for the average value of $\mathrm{\mid }{\mathrm{C}\mathrm{l}}_{\text{Ł}}^2\mathrm{/}{\mathrm{C}\mathrm{l}}_{\text{Ł}}^4{\mathrm{\mid }}^m$, where $\text{Ł}$ varies over an infinite family of quadratic extensions of certain Galois number fields. As a special case of our theorem, we obtain lower bounds for the average value when the base field is any Galois number field with class number $1$ in which $2\mathbb{Z}$ splits.