Hey, @reanor.
As Marko said, this is not really something we’ll help with. It’s completely normal gameplay.
If it’s a rush, you can just replace the cores, and then they’ll be private to yours. Just keep in mind that Alien Cores and Patreon Cores can’t be picked up with a multitool.
https://forum.empyrion-homeworld.net/t/read-first-will-you-receive-support-yes-or-no/72