I'm being pedantic here because this is my pet peeve. Allowed is the correct word.
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For the swinger sex to happen, Nick Rekieta must have told or
VERY STRONGLY implied to Kayla Rekieta that:
-She would not get in trouble for fucking Aaron
-She has permission to fuck Aaron
-He would not prevent the Aaron-fucking
Ordinary dating and/or marriage partners do not simply fuck anyone they come across and expect it to have no consequences for their relationship. Why is that? In my opinion, the logical conclusion is that they do feel a claim of ownership from their significant other. If Kayla isn't Nick's wife, then she becomes "our" communal wife, or FapCop's wife, as people here joke frequently.
Someone owns a wife, and someone owns a husband. The question is, who?