A measurement of the CKM angle $\gamma$ is performed in $B^{\pm} \to D K^*(892)^{\pm}$ decays at the LHCb experiment, where $D$ represents a superposition of $D^0$ and $\overline{D}{}^0$ states. Using the dataset collected during Run 1 and Run 2, this analysis represents a comprehensive study of this channel, with the $D$ meson reconstructed in two-body final states $K^{\pm}\pi^{\mp}$, $K^+K^-$ and $\pi^+\pi^-$; four-body final states $K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$ and $\pi^+\pi^-\pi^+\pi^-$; and three-body final states $K^0_{S} \pi^+\pi^-$ and $K^0_{S} K^+ K^-$. This measurement constitutes the first observation of the suppressed $B^{\pm} \to [\pi^+K^-]_D K^{*\pm}$ and $B^{\pm} \to [\pi^+K^-\pi^+\pi^-]_D K^{*\pm}$ decays. The combined result gives $\gamma=(63\pm 13)^\circ$.