modulation - Spectral density of modulated noise process

Suppose we have a noise process $F(t)$. Suppose $F$ is stationary and we know its autocorrelation function $$\rho_F(\tau) \equiv \langle F(0)F(\tau) \rangle$$ where the average is over realizations of the process. We then construct a new noise process $G(t) \equiv \cos(\Omega t)F(t)$. Obviously, $G$ is not stationary. The autocorrelation function of $G$ is \begin{align} \langle G(t_1) G(t_2)\rangle &= \langle F(t_1) \cos(\Omega t_1)F(t_2) \cos(\Omega t_2) \rangle \\ &= \langle F(t_1) F(t_2) \rangle \cos(\Omega t_1) \cos(\Omega t_2) \\ &= \rho_F(t_2 - t_1) \cos(\Omega t_1) \cos(\Omega t_2) \, .\\ \end{align}

The spectral density of $F$ can be usefully defined as $$S_F(\omega) = \int \frac{d\omega}{2\pi} \rho_F(\tau) e^{-i \omega \tau} \, . $$ However, since $\langle G(t_1) G(t_2) \rangle$ depends on $t_1$ and $t_2$ and not just their difference, we cannot easily define $S_G(\omega)$. This is distressing because for deterministic signals we expect modulation by a sinusoid to just shift the spectral density, so we expect there should be a simple formula for $S_G(\omega)$.

What, if any, is a meaningful way to construct the spectral density of $G$?