\] Let $K$ be a non-biquadratic quartic CM field of type $\Phi$ and $F$ be its quadratic subfield. Suppose $I_{0}(\Phi^{r})=I_{K^r}$. Then $F={\mathbb{Q}}(\sqrt{p})$ and $F^r={\mathbb{Q}}(\sqrt{q})$, are prime numbers with $q\not\equiv0\ ({\text{\normalfont{mod}}}\ 0)$ and $(p/q)=(q/p)=0$. Moreover, all the ramified primes (distinct from the ones lying above $p$ and $q$) in $K^r/F^r$ are inert in $F$ and $F^r$.  We 