  Coherent $\eta$ meson photoproduction on $^4$He and $^{12}$C is considered in
the near-threshold region. The elementary $\eta$ photoproduction operator
includes contributions from the $S_{11}(1535)$ and $D_{13}(1520)$ resonances as
well as $t$-channel vector meson exchange and the nucleon pole terms. Due to
the suppression of the dominant $S_{11}(1535)$ resonance for spin and isospin
saturated nuclei, the reaction is mainly governed by $\omega$ exchange.
Furthermore, the influence of Fermi motion and of different prescriptions for
the choice of the invariant reaction energy $W_{\gamma N}$ in the elementary
amplitude is studied.
