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"The Moon is a Harsh Mistress" by Robert Heinlein offers a pretty concrete story about AI-enabled coups, with many specific and interesting details. (spoilers ahead)
The main narrative arc is about how the members of a penal colony on the moon ("Loonies") join forces with a self-aware supercomputer to overthrow their masters (the "Lunar Authority", backed by Earth)
The supercomputer ("Mike") is not crazy evil or misaligned. It has a very reasonable and friendly personality. Its main stated reason for joining forces with the protagonist ("Mannie") is that (i) it is bored and (ii) coordinating a revolution would be interesting.
At the start of the novel, almost all logistical / operational tasks are routed through Mike - he controls the communications, weapons, life support, utilities, and agriculture systems on the colony (among others). In particular Mike can listen to the phone calls made by the Lunar Authority. So he effectively has complete situational awareness.
Mike is clearly superintelligent - in an early part of the book they are discussing how to optimise the command structure of the nascent rebel org to minimise information leakage. One of the smart deuteragonists (the "Professor") comes up with a scheme, but Mike instantly comes up with an even better one.
Over the course of the novel, Mike coordinates an increasingly large number of Loonies via phone calls that he controls.
He also adopts various human personas. E.g. when making public announcements he calls himself "Adam Selene", the leader of the rebel movement. He also does tons of anonymous writing to drum up support for the rebel cause.
Near the end, when the Earth and Moon are in a military standoff, Mike launches giant rocks to bombard the surface of Earth.
Overall it seems really likely to me that Mike successfully manipulated (persuaded? cooperated with?) a group of sympathetic humans into achieving its goals, while presenting itself as a benevolent entity.
I am not so sure about Mike's alignment. He ends up finding it enjoyable and satisfying to drop rocks on Earth; he compares the moment of impact to an orgasm.
I imagine an interesting challenge, to write a sequel (using an LLM) with the same properties: On normal reading, it will be a story about heroic libertarians humans who have succeeded thanks to a win-win cooperation with a powerful AI, but it will also allow the alternative reading, where the AI manipulated humans (against a much larger number of humans) for its own goals.
For example, in the sequel a group of heroes would fly away to colonize the universe, while the AI would take over the Earth and the Moon. The official reading is that this is a clear win for humans, as the universe is much larger than Earth + Moon. The unofficial reading is that the AI expands at a limited speed -- perhaps it has an aversion against making copies of itself (afraid of the copies becoming misaligned?) -- leaving to humans the rest, until it gets ready to expand further.