To follow up from my earlier report I checked today on my first system where I built an installation yesterday to see if the colony economy of the station changed - it didn't. The population changed so the installation finished having an effect overnight but didn't cause a recalculation of the station economy. That's not unexpected because the installation was built around another body and it has no economic influence either (comms). It's a shame since the outpost is around a rocky ice body so it would've add some good info.
I then decided to use my test system to build a commercial outpost around the star to see what happens. This was quite a bit more interesting. The system is around a class Y brown dwarf star. For the primary port I had previously built a military outpost which is around an ice moon. I tried to place the new construction in slot 1 but the game felt I meant slot 0 and put it there. Building things was ... different. The construction was 0.16 Ls away from the jump in point so on arrival I had to go to 0% power to line up or loop around (very slow in a T9). On leaving jump I had to use a heatsink (or burn up at 130+%) since it was so close to the star.
Roughly 5 hours later ... finished the construction and the economy is military (proportion 1.0). The existing military outpost didn't change. The new one sells the exact same things and amounts as the existing military outpost. I briefly got paranoid I built the wrong thing but using the rename inspection trick the icon was for commercial so it just picked up the military influence.
With only a single data point it's hard to tell but I'm thinking outposts around a star pick up the system economy as the influence. It could be that stars are just military but that seems a little odd to me (subjectively, no real basis). From here I think I might finish my Coriolis in my first system. It is around the star too and currently the system economy is colony. That may tell us more. It will take a small bit though as I probably have about 13 hours of hauling for it left.
I completed a Commercial Outpost (Plutus) at the main star in a new empty system, before the Thursday downtime. It came online as a 'Colony' economy. Then, yesterday I added a Mining/Industrial Installation (Phorcys) at the same star and the outpost changed to 'Extraction/Military', with the system in the Galaxy Map now showing as 'Military/Extraction'.
Outpost reports: Military 1.0, Extraction 0.3
Outpost commodity market reads 'Military' but the goods sold just looks like Extraction (although, it's a Patronage faction so they may be suppressing any military goods).