I just finished the book- I forgot that most of the battle is actually described in First Strike, after the MC gets back to Reach (yay time distortion). However, of the PoA I can say that it's en route to the ONI mission to kidnap a prophet when Reach is attacked. It returns to orbit in time to launch a two-pronged assault with the Spartans on board. The MC, Linda, and James breach the orbital dock to erase some info on Earth's whereabouts, where they meet Johnson and his team. All of them then return to the Autumn, which escapes to Halo. In the process all the Autumn's uber-ONI-gear is destroyed, leading to its pitiful defense and subsequent crash on Alpha Halo's surface.

All the other Spartans on the Autumn head to the surface, where First Strike picks up later. If any events from the books are to be seen in Halo: Reach, it would probably be that ground assault, or maybe some transmissions from orbit. The Autumn itself was never really involved.