= Solution
In Questions 3–4, use primes for conformal-time derivatives and $\mathcal H=a'/a$; proper-time derivatives will be stated separately. Let $\xi^\mu=(T,L^i)$. The metric is a <covariant tensor>, so under the passive coordinate change $\widetilde x^\mu=x^\mu+\xi^\mu$,
$$
\widetilde g_{\mu\nu}(\widetilde x)=
\frac{\partial x^\alpha}{\partial\widetilde x^\mu}
\frac{\partial x^\beta}{\partial\widetilde x^\nu}
g_{\alpha\beta}(x).
$$
The inverse Jacobian is $\delta^\alpha_\mu-\partial_\mu\xi^\alpha$ to first order. At the same new coordinate argument, the homogeneous background satisfies $\bar g_{\alpha\beta}(x)=\bar g_{\alpha\beta}(\widetilde x)-T\bar g'_{\alpha\beta}$. Shifting the argument of $\delta g$ or multiplying it by a Jacobian perturbation produces only second-order terms, which we discard. Subtracting the unperturbed metric therefore gives
$$
\boxed{\delta\widetilde g_{\mu\nu}=\delta g_{\mu\nu}
-\bar g_{\alpha\nu}\partial_\mu\xi^\alpha
-\bar g_{\mu\beta}\partial_\nu\xi^\beta
-T\bar g'_{\mu\nu}}.
$$
This is the printed Jacobian expression expanded to linear order, equivalently $\delta\widetilde g=\delta g-\mathcal L_\xi\bar g$. It is a <gauge transformation in cosmological perturbation theory>, not a change of the physical spacetime.
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