I'm trying to explain the behaviour of a geostationary satellite using different frames of reference.
Inertial frame: The satellite has a circular motion with angular velocity $\omega$. The centripetal force $F$ required for this motion is created by the gravitational pull of Earth. Earth itself rotates around its axis with $\omega$, but that is irrelevant. OK
Rotating frame ($\omega$): The frame of reference is fixed to Earth. Everything appears stationary. Gravity is still present, which still acts on the satellite with force $F$. Due to the acceleration of our frame of reference we introduce a centrifugal force, which acts on the satellite with $-F$. The forces cancel out, so the satellite's lack of acceleration is explained. OK
Rotating frame ($2\omega$): This frame of reference rotates around Earth's axis with angular velocity $2\omega$. The satellite appears to have angular velocity $-\omega$. The centripetal force $F$ is provided by gravity. However, we have not yet accounted for the acceleration of our frame of reference! There should be a centrifugal force of $-2F$, meaning that the satellite should be accelerating away from Earth!
How do we explain case 3?