That’s right, four currently acceptable spellings. That’s nothing really, the word has gone through 14 different spellings over the centuries. I find this odd, particularly that the spelling of this word still hasn’t been nailed down by this point. Long may it continue, however, I’d hate to see some committee try to impose a Standard English on us. There is a “Standard English”, of course, but it is a living thing, slowly shifting over time and not controlled by some faceless central authority.
aerie, aery, eyrie, eyry
1. The nest of any bird of prey; especially, in modern usage, of an eagle; also extended to that of ravens and other birds building high in the air; and fig. to a human residence or retreat perched high on a rock or mountain side.
2. The brood in the nest; the young of a bird of prey, or fig. a noble stock of children.