So this excellent post a week ago by /u/idjet explains the circumstances of the passing on of Charlemagne's imperial title, as well as whether he intended to do so or not.

However, whatever the details were, after the division of the empire after the death of Charlemagne's surviving son Louis the Pious, the title was passed to Louis' son Lothair because in the triparte split of the empire, Lothair possessed the core Frankish lands in middle francia as well as Italy and Rome. And since it was understood that in order to be emperor, one needed to be in possession of Rome, Lothair was thus designated the emperor.

This continued on after his death with his son Louis II, who was emperor even though he possessed no part of the former frankish homelands, and only retained the title King of Italy. Though the empire, the imperial title, and the frankish royal title would be reunited again under Charles the Bald and Charles the Fat, after the latter's deposition, you would see a similar situation as to what happened with Charlemagne. The various component kingdoms of the Carolingian Empire would be divied up (although this time the inheritors would not necessarily be Carolingians), and it only took someone in possession of the title of King of Italy to be crowned emperor (although not all kings of italy would be crowned emperor). And they would only be crowned because there was a need for legitimation or other political concerns, local entirely to Italy, as opposed to the rest of the former Carolingian Empire.

Which is how, the titles of "King of the Franks" and its successor titles, King of the West Franks (West Francia), Middle Franks/Francia (Lotharingia/Lorraine) and East Franks/Francia came to be split from the imperial title.