= Solution
Identify $G\simeq(\mathbb Z/15\mathbb Z)^\times\simeq C_4\times C_2$, with $\sigma(\zeta)=\zeta^2$ and $\tau(\zeta)=\zeta^{-1}$. Its <subgroups> are
$$
1;\ \langle\sigma^2\rangle,\langle\tau\rangle,\langle\sigma^2\tau\rangle;\ \langle\sigma\rangle,\langle\sigma\tau\rangle,\langle\sigma^2,\tau\rangle;\ G.
$$
Their fixed fields, in the same layers, are
$$
L;\ \mathbb Q(\sqrt{-3},\sqrt5),\ L^+=\mathbb Q(\zeta+\zeta^{-1}),\ \mathbb Q(\zeta_5);\ \mathbb Q(\sqrt{-15}),\mathbb Q(\sqrt{-3}),\mathbb Q(\sqrt5);\ \mathbb Q.
$$
The real fields are $\mathbb Q$, $\mathbb Q(\sqrt5)$, and $L^+$. This is the <Galois correspondence>.
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