Charterhouse is the name of one of the great public schools of London.
But whether or not the institution has a charter, the fact has no bearing on its name.
Its site was originally that of a Carthusian monastery, an order which derives its name from that of the French village where it first arose.
That village was La Grande Chartreuse.
Its early English spelling was Chartrous.
Combined with the fact that the monastery then bore the French name, maison Chartreuse, “Carthusian house,” the Londoners quite naturally amplified Chartrouse into Charterhouse.