We have a tower of normal extensions of $\mathbb Q$:
$$ \mathbb Q \subset M \subset \mathbb Q(\zeta_m).$$
In this situation, we have a surjective group homomorphism:
$$ \phi : Gal(\mathbb Q(\zeta_m) : \mathbb Q) \to Gal(M : \mathbb Q),$$
which sends each $\sigma \in Gal(\mathbb Q(\zeta_m) : \mathbb Q)$ to its restriction $\sigma|_M \in Gal(M : \mathbb Q)$.
[Since $M : \mathbb Q$ is a normal extension, we have $\sigma(M) = M $, so this definition makes sense. And since $\mathbb Q(\zeta_m) : \mathbb Q$ is a normal extension, every $\mathbb Q$-automorphism of $M$ extends to a $\mathbb Q$-automorphism of $\mathbb Q(\zeta_m)$, which means that $\phi$ is surjective.]
Anyway, the kernel of the group homomorphism $\phi$ is $Gal(\mathbb Q(\zeta_m) : M)$. Hence $\phi$ induces a group isomorphism:
$$ \frac{Gal(\mathbb Q(\zeta_m) : \mathbb Q)}{Gal(\mathbb Q(\zeta_m) : M)} \cong Gal(M : \mathbb Q),$$
which is the isomorphism you are looking for.
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