Cyclicity of the 2-class group of the first Hilbert 2-class field of some number fields

Let $\text{mathds}k$ be a real quadratic number field. Denote by ${\mathrm{C}\mathrm{l}}_2(\text{mathds}k)$ its $2$-class group and by $\text{mathds}{k}_2^{(1)}$ (resp. $\text{mathds}{k}_2^{(2)}$) its first (resp. second) Hilbert $2$-class field. The aim of this paper is to study, for a real quadratic number field whose discriminant is divisible by one prime number congruent to $3$ modulo 4, the metacyclicity of $G=\mathrm{G}\mathrm{a}\mathrm{l}(\text{mathds}{k}_2^{(2)}\mathrm{/}\text{mathds}k)$ and the cyclicity of $\mathrm{G}\mathrm{a}\mathrm{l}(\text{mathds}{k}_2^{(2)}\mathrm{/}\text{mathds}{k}_2^{(1)})$ whenever the rank of ${\mathrm{C}\mathrm{l}}_2(\text{mathds}k)$ is $2$, and the $4$-rank of ${\mathrm{C}\mathrm{l}}_2(\text{mathds}k)$ is $1$.