Corollary 8.7 in Cox's primes of the form $x^2+ny^2$: given a number field $K$ and two abelian extensions $L, M$, $L\subseteq M$ if and only if there is a modulus $\mathfrak m$ divisible by all primes of $K$ that ramify in either $L$ or $M$, such that $$P_{K}(\mathfrak m)\subseteq \ker\Phi_{M/K,\mathfrak m}\subseteq \ker\Phi_{L/K,\mathfrak m}$$ where $\Phi_{L/K, \mathfrak m}$ is the Artin map $\left(\frac{L/K}{\cdot}\right):I_{K}(\mathfrak m)\to Gal(L/K)$ on the ray class group defined by the modulus $\mathfrak m$.
I understand the proof of the forward direction, but I can't see why Cox's proof of the converse makes sense. Specifically,
If there is a modulus $\mathfrak m$ such that the inclusions hold, then the kernel of $\Phi_{L/K, \mathfrak m}$ is mapped to a subgroup of $Gal(M/K)$ by the map $\Phi_{M/K,\mathfrak m}$, which by Galois correspondence, there is an intermediate field $M/\tilde L/K$. But then applying the first part [the forward direction] to the extension $M/\tilde L$, we get $\ker\Phi_{L/K,\mathfrak m}=\ker\Phi_{\tilde L/K,\mathfrak m}$.
I could not see why the two kernels are the same. Indeed, I understand that $I_K/\ker\Phi_{\tilde L}\cong H=\ker\Phi_L/\ker\Phi_M$ and that $\ker\Phi_M\subseteq \ker\Phi_{\tilde L}$ but I couldn't see how to relate $L$ and $\tilde L$. Any help is appreciated. Thanks in advance.