question to CORE leaders.

[Robert.Fitzgerald]

The Core is a militaristic group, unlike the quite rag-tag Hunters section of the Guild. It is unlikely that capital ships could be commanded in such harsh environments by individualistic "normal" bounty hunters. I presume the Core pilots are the best of the best, and have extra training in areas of leadership, discipline etc.

The Core's goal is to harvest nomads for their technology, for an unknown future purpose (war? upgrading their hunters? etc). The Core encountered another human group with a similar purpose, the Order, and a conflict ensued. The Order and the Guild Core have been fighting in the Omicrons for years, both sides trying to oust the other to lay claim to the nomad technology.
The Corsairs have made their presence known in the Core's area of operations (a glaring example would be Yaren Base in Omicron Delta). The Corsairs support the Order both financially and in terms of manpower and technology, so they are a (secondary) target of the Core.
About the "who supports the Core" question, the Core is funded largely from the profits of the Hunters section of the Guild. According to the capital ship infocards, the Liberty government assisted the Guild Core in the development and early construction of said ships. The other Houses don't seem to be related to the Guild Core.

The Guild Core would not necessarily encounter the Outcasts often, as their main areas of operations are in the nomad systems not in Omicrons/Taus. However, if a Guild Core pilot learned about Outcast-nomad worship I'm fairly sure he/she wouldn't be so chummy with Outcasts.
Corsairs are higher on the Guild Core's priority list of targets, as they directly support the Order and impede the Core operations via raids and assaults.

The Guild Core has no actual bases, unless you count Rostock and the Omega 56 bases as Core facilities. This might be due to Igiss' not including them accidently or on purpose. I would expect the Guild Core ships to operate out of Omega 15-56 and Omicron Delta (the Freeport). As most of the ships are capital class and/or designed to work in harsh environments, I would say (in-RP) that Guild Core forces would mostly survive without relying on stationary bases, by hiding in dust clouds / nebulas and relying on Zoner supplies.
^
All of that I have gleamed from what little information there is on the Guild Core. I have one Guild Core independent character, who flies an Orca.

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