The rules are messy, but all I see as mandatory is that you declare your detachment either <Kabal>, <Wych cult>, or <Coven>, and then only those units gain an obsession. That's it, there are no restrictions to which units you can include in that detachment.
So you can include a Cronos in a <Wych Cult> detachment, but it does not get any Coven obsession.
You could even have a <Coven> detachment include a <Cult of Strife> Succubus as your warlord, who would not get the obsession, but could get the warlord traits and relics from the Charadon book (but not the stratagems, as that requires a <Cult of Strife> detachment).
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