Despite being aligned with the New Monarchy, I could see this as a possible thing. They will do whatever it takes to preserve Earth with a strong government, ruled by an iron fist.
If there were faction versus faction matches in the Crucible, I will bet it's not for the sake of battle, but to promote themselves and their cause to get other Guardians to join. There's probably way more groups than just the main three we know right now.
Edit: A word
English
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There was originally supposed to be 7 factions
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I think much the same. I hope we see more of these groups in future releases, starting with Destiny 2.