What happens to auras that come back from exile?
The comprehensive rules say this about Auras as they enter play:


  303.4. Some enchantments have the subtype “Aura.” An Aura enters the battlefield attached to an object or player. What an Aura can be attached to is defined by its enchant keyword ability (see rule 702.5, “Enchant”). Other effects can limit what a permanent can be enchanted by.


So, no matter how an aura enters the battlefield (usually by casting it, but Synod Sanctum works too) you choose what it is attached to. This is further specified in sub-section f:


  303.4f If an Aura is entering the battlefield under a player’s control by any means other than by resolving as an Aura spell, and the effect putting it onto the battlefield doesn’t specify the object or player the Aura will enchant, that player chooses what it will enchant as the Aura enters the battlefield. The player must choose a legal object or player according to the Aura’s enchant ability and any other applicable effects.