The short version of the faction is they're a military junta born out of a crisis in the home system, when an anarchy faction almost toppled the incumbent Imperial Patronage. The story goes that the patronage declared martial law and ceded control of the system to this faction[1], who proceeded to quash the pirate faction[2] and restore order. So impressed by this action, the people of the system refused to return to their previous rulers[3]. This all happened under the watch of an Imperial power, which boded well for their ultra-nationalistic views. Essentially, while they reject Federal rule entirely, they also reject homeland Imperial authorities, believing themselves to be practitioners of "true imperial values". As such, entities like Nova Imperium saw significant support from representatives[4] of the faction, and the faction would cooperate with Alliance and Independents in the region, before cooperating with homeland Imperials.
Meanwhile, our group are a shadow "loyalist" group conducting research into Thargoids, Guardians and other alien entities, in order to develop a military edge for the Empire. A clandestine relationship between the group and the faction exists, whereby the faction accepts assistance with "less savoury activities" which may reflect badly on the faction proper. It's not unusual for our members to be unfriendly/hostile to most other factions except our own, except where it's explicitly to our advantage. In exchange, they're our "test grounds" for this research, whether on their own populace or against it's enemies. They know a good deal of their successful expansion into many systems is from our own efforts, so they know better than to question when we depart to conduct investigations or return with alien samples. Obviously we view Canonn, Ram Tah and Palin as highly beneficial relations, and we'll always participate in operations related to alien activity. Ultimately, these acitivites are for the betterment of the Empire, rather than any other altruistic goals like some of the other research entities have in mind.
.... that's the short version, but it's two very distinct profiles, and it means anyone dealing with us gets a very different read on things compared to "with the faction"; the faction hates Feds, but the group doesn't necessarily. The faction (from our group's perspective) was never actually meant to grow, but it just kinda did. We went into this with no delusions of grandeur and, as a small group, have 100% expectation for the group to live and die by the actions of others.
[1] Actually happened; there was an issue with the BGS and influence swings were massive, turning a 50% vs 10% difference to something like 30% vs 25%. As the faction was in the first tranche when FD were a bit more liberal with putting PMFs in the game, they were given control of the system and a couple assets immediately, but with only 10-15% influence. Several wars later and they were comfortably at 50-60% with a few more assets to their name
[2] I would've liked to have had a "morally questionable" conflict CG to exterminate the pirate faction resulting in them being removed from the game, with the alternative being they take control of the capital. Unfortunately, not doable.
[3] At the time, my group had forgotten about the PMF submission, so we'd grown the incumbent ruler to a half dozen systems. When they came about I asked the group if we wanted to support both groups, and just avoid our faction conquering the old rulers. They declined, and we've pretty much knocked them out of any form of control now
[4] Those representatives being the group, not the PMF.