So you also have an intermittent no start condition. That definitely makes the cheese more binding. I don't know what a GM extended warranty goes for but you may want to consider it unless you're really at least a shade tree mechanic or know one. And you will absolutely need the service manual (paper about $150; DVD about $400). Just sayin'.
All or some of the issues you are having may be connected. I see you are adding posts while I compose this so some of this might not apply.
The possibilities for setting the P0650 are:
All apply to the MIL circuit:
ECM is bad-possibly a high resistance connector issue (may be as simple as cleaning & reseating the connector)
IPC is bad-possibly a high resistance connector issue
The ground control circuit wire is open or shorted
The MIL ignition voltage circuit is open or shorted
So what it amounts to is you can't rely on anything your MIL does right now.
From the '06 G6 SM:
Turn OFF the ignition.
Disconnect the engine control module (ECM).
Turn ON the ignition with the engine OFF.
⇒ If the MIL is still ON, test the MIL control circuit for a short to ground.
⇒ If the MIL control circuit tested OK and the MIL stayed ON, replace the IPC.
⇒ If the MIL control circuit tested OK and the MIL went out when the ECM was disconnected, replace the ECM.
Measure for 12 volts from the MIL ignition voltage circuit in the IPC harness connector to a ground.
⇒ If there is less than 12 volts, test the MIL ignition voltage circuit for an open, or a short to ground, and an open fuse.
Remove the fuse that supplies voltage to the MIL.
Measure for less than 1 volt from the MIL control circuit in the ECM harness connector to a good ground.
⇒ If there is more than 1 volt, test the MIL control circuit for a short to voltage.
Install the fuse that supplies voltage to the MIL.
The MIL should illuminate with a 3-amp fused jumper wire connected between the MIL control circuit in the ECM harness connector and a good ground.
⇒ If the MIL does not illuminate, test the MIL control circuit for an open or high resistance.
⇒ If the MIL control circuit tests OK, replace the IPC.
⇒ If the MIL does illuminate, but does not when commanded ON with a scan tool, replace the ECM.