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For the purposes of this question, let's say Charlemange was not excited by the prospect of being crowned titular emperor of a dead, non-frankish empire and decides to have the Pope crown a male from one of the few remaining Roman aristocratic lines.
Charlemange of course retains all the power as king, with the potential HRE acting as a puppet figure head for him to control.
How plausible is this, and what are the ramifications if it were to happen this way?