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When starting the car, cranking the starter, is it normal operation for the stereo to lose power and come back after the start?.. Because that's what's been happening. The stereo is not stock, alpine deck, ground/ignition/power. Power is wired to the cigarette lighter hot wire.
I've also seen a video of another later model c3s when starting their power antenna will raise and lower then back up on startup..
IIRC when it goes to the crank position, all of the other circuit that are controlled by the ignition switch (wipers, etc) are turned off while those that don't get affected by the ignition switch (headlights) of course stay on.
When the starter is cranking and drawing a lot of power, if your battery is weak(ish) and the power available to the stereo drops below whatever its threshold requirements are (might be 10v) it will turn it's self off until the power gets up past what it needs again.
Most modern stereos require a constant power and a switched power. Your switched power seems to be correctly hooked up to acc power, because it is off during cranking.