= Solution
For each $k$, the lifted <Riemannian metric> on $X$ is $T$-invariant and the two <subgroups> still act freely. <Sunada's theorem> therefore gives
$$
\boxed{\operatorname{Spec}(N,g_{1k})=\operatorname{Spec}(N,g_{2k}).}
$$
To rule out any other equalities, the leading diagonal <heat kernel expansion> from Question 3 gives, on a closed surface,
$$
\operatorname{Tr}(e^{-tP})=\frac{\operatorname{area}(N)}{4\pi t}+O(1)\quad(t\downarrow0).
$$
Thus the function <spectrum> determines area: multiply the <heat trace> by $4\pi t$ and take the limit. Since the chosen areas are distinct for distinct $k$, no cross-pair spectra can agree. This proves exactly the prescribed spectral coincidences.
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