@Jockgit64
To change control of a station, you need a civil war happening between the owner and whomever you wish to take control.
So do missions for Imperial Society, do bad things to the owners of the platform, and also mess up their missions. I think the key was to get their Civil Unrest up. (What I don't know is if there's a specific subset of missions or things you should do to make this happen faster and cause a Civil War).
The system itself is Empire aligned because the 'main station' is owned by an Empire faction. So say the Mobius Platformers won the civil war, they might end up flipping a station to their side and if it's the main one then the whole system goes to them.
System-wide influence is 100% distributed between all factions (it is rounded, so it doesn't show the decimal places, that might be your missing %'s). Factions gain influence from trade with their controlled station, and mission successes.
So basically things that you can do to support the Empire:
- Trade with the Empire faction
- Succeed at missions in the Empire faction
- Destroy enemy faction ships (e.g. wanted ships, but make sure you process the bounty at an Empire station)
- Sabotaging any missions you might find from the enemy faction. You can do this by failing them intentionally, by abandoning them, or accepting a counter offer at a USS.
That's all I've learned so far watching both Lugh and Mikunn threads
There's probably other things I haven't seen/picked up yet though!