When Flicker of Fate exiles a permanent, that permanent leaves the battlefield. Any Aura enchanting it immediately loses its legal enchantment target, since the enchanted object no longer exists on the battlefield.
State-based actions are checked continuously. CR 704.5m states that if an Aura is attached to an illegal object or is not attached to anything it should be, it is put into its owner's graveyard. Because the enchanted permanent has left the battlefield, the Aura has no legal object to remain attached to, so it is placed into its owner's graveyard the next time state-based actions are checked.
When the exiled permanent returns to the battlefield (as a new object under Flicker of Fate's effect), the Aura is already gone. The returning permanent comes back free of the Aura, and the Aura stays in the graveyard.
Example: Your opponent controls a Grizzly Bears enchanted with Arrest. You cast Flicker of Fate targeting Grizzly Bears. Grizzly Bears is exiled, Arrest has no legal permanent to be attached to, and state-based actions put Arrest into the graveyard. Grizzly Bears then returns to the battlefield untapped and free of Arrest.
Note: This is different from Equipment, which simply becomes unattached and stays on the battlefield (CR 704.5p). Auras must always be attached to a legal object or they go to the graveyard (CR 704.5m).
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