Gerth generalised Cohen–Lenstra heuristics to the prime \(p=2\) . He conjectured that for any positive integer m, the limit \( \lim _{X \rightarrow \infty } \frac{\sum _{\begin{array}{c} 0< D \le X, \\ { \text {squarefree} } \end{array}} |\textrm{Cl}^2_{{\mathbb {Q}}(\sqrt{D})}/\textrm{Cl}^4_{{\mathbb {Q}}(\sqrt{D})}|^m}{\sum _{\begin{array}{c} 0 < D \le X, \\ { \text {squarefree} } \end{array}} 1} \) 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 \(|\textrm{Cl}^2_{{\textbf{L}}}/\textrm{Cl}^4_{{\textbf{L}}}|^m\) , where \({\textbf{L}}\) 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 of \(|\textrm{Cl}^2_{{\textbf{L}}}/\textrm{Cl}^4_{{\textbf{L}}}|^m\) as we vary \({\textbf{L}}\) in an infinite family of quadratic extensions of certain Galois number fields of class number 1 containing \({\mathbb {Q}}(i)\) .
