The Charlemagne who appears in the legendary Matter of France is not always on the side of the heroes, and there are those who stand opposed to him - not only the villainous traitor Ganelon, but heroic characters as well who, at least at times, stand in opposition to the king. They are the central figures of their own sub-cycle within the legends, recorded in folktales as often as in literary chansons de geste. Notably, though, folklore remembers these people more for their horse than for their specific acts of defiance.
The literary chansons attribute a lot of the stories of those who stand against Charlemagne to descendants of one Doon de Mayence, whose twelve sons include Aymon of Dordone as well as the fathers of both Ganelon and Ogier the Dane. Both Ganelon and Ogier are part of the familiar stories of the Twelve Peers, but Aymon, his children, and their legendary horse Bayard belong to a distinct branch of the tradition which survived in the folklore of the Low Countries into the modern era.
It is the chansons which give us the oldest known version of the story, in which Aymon has four sons, of whom he favors the eldest, Renaud. Renaud gets into a brawl with a close relative of Charlemagne (in the earliest texts his nephew, while other versions make it his own son); they flee and go into hiding, with the help of their horse, Bayard, which is so large it can carry all four sons on its back at once. Some number of adventures follow until ultimately the sons are besieged by Charlemagne's forces at the castle of Montauban, where they are defeated. Charlemagne orders the horse to be drowned and commands that Renaud go on crusade as penance (in texts not dating to the crusades, he simply goes on a pilgrimage to Jerusalem); when he returns, he becomes a common laborer, helping to build a church. He is then murdered and his body thrown into the river.
The horse is remembered more than the specific adventures of any rider, at least in modern folklore. It's specifically associated with certain towns in the Low Countries, where it's sometimes said to have been able to change its size at will to accommodate its rider(s), to have understood human speech or even had the power to speak itself. Most tellings include a detail found in a few manuscripts of the chansons de geste, that Charlemagne ordered that the horse be drowned by tying a large rock to it; Bayard splits the rock with his great hooves and runs away to live wild in the woods. The rock is sometimes said to be a particular split rock which is part of the landscape near Dinant, Belgium. The horse is sometimes said to have been given to the family by a fairy or, more commonly, by the sorcerer Maugis (who is himself a descendant of Doon de Mayence, if the literary version of the tradition is to be believed).