Autocovariance of ARMA-GARCH vs. that of pure ARMA Is the autocovariance of an ARMA-GARCH process the same as that of the ARMA part of that process?
If this is too difficult/cumbersome to show, analysis of a special case like MA(1)-GARCH(1,1) or AR(1)-GARCH(1,1) would still be helpful.

If $\epsilon_t$ is a GARCH process where the conditional variance is given by
$$
h_t=\alpha_0+ A(L)\epsilon_t^2 + B(L)h_t
$$
then $\epsilon_t$ is wide-sense stationary with $E(\epsilon_t)=0$, $\operatorname{Var}(\epsilon_t)=\alpha_0(1-A(1)-B(1))^{-1}$, and $\operatorname{Cov}(\epsilon_t,\epsilon_s)=0$ for $t\neq s$ if and only if $A(1)+B(1)<1$ (see Bollerslev 1986, Theorem 1), that is, $\epsilon_t$ is uncorrelated (but not necessarily iid) white noise.
Hence, if $X_t$ is an ARMA process satisfying $\phi(L)X_t =\theta(L)\epsilon_t$, its autocovariance function follows in the usual way (e.g. Brockwell & Davis, 1991, § 3.2).
