Well now you need to take over the station (I won't mention the location of your secret base). You only have the one base so far, yes? Duck was there last night - amazing view! I think I'll send Jenny over to work on your influence, as that's more her region of space than Duck's.
Duck is getting the itch to go on a short but potentially profitable exploration run, and if you can take 'own' that station, I'll be able to cash in my cartographic data there. I believe that aids the controlling faction, right?
UPDATE - Jenny has moved much of her fleet to [REDACTED]. She flies in Open, so maybe you'll see her. LMK what is the best way to push a faction to control a system, and I'll focus on that.