I treat the Ine Givar as a rough equivalent of Communist parties. Some groups engage in armed struggles of various kinds. Others are able to engage in democratic politics, even win elections, and occasionally not be overthrown.
The Imperium is an oligarchy, economically dominated by the megacorps. In addition, there is widespread and systematic speciesism, with groups like the Vargr and the "Dandies" of Junidy routinely facing various forms of oppression. In addition, of course, you can be lynched or lobotomised simply for developing your natural psionic talents...
Opposition to one or most aspects of this is inevitable. The Ine Givar are merely the most radical expression of this opposition.
They are, however, extremely factionalised. This has gotten worse since the Fifth Frontier War. Many groups took up arms in support of the Zhodani during this war, and lost heavily. Other groups refused to do so, effectively swearing allegiance to the Imperium.
There are also cultural differences. IMTU, many of the Imperial worlds near the Sword Worlds are culturally very similar to the Sword Worlds. In fact, they were settled at the same time as the Sword Worlds, by the same people. Subsequent Imperial colonisation has diluted this a little, but hasn't essentially changed the fact that these are more or less Sword World/Solomani inhabited worlds. Similarly, there are many Vargr on other worlds. That status of Vargr on these worlds is in many cases an important political issue. The Ine Givar often has considerable influence amongst Vargr and their sympathisers on these worlds. (Actually, they mainly work through alliances with Vargr groups, because of charisma issues, but that's another story...)
Other groups are heavily oriented towards psionics' rights, and are (usually) pro-Zhodani. A few groups look to Arden as something of a model. Other groups are rather isolationist.
Various intelligence services are involved with them at one level or another. The Imperium has found them difficult to infiltrate, but seems to have done so on occasion. At least a few of the IG's most notorious actions were undertaken by Imperial intelligence agents, seeking to discredit the organisation. Other agents probably attempt to influence individual groups in more moderate directions. The Zhodani cooperate with them, of course. They supply training and provide material assistance to approved groups, and generally attempt to sway the IG's agenda in their direction. The Sword Worlds, Arden, and various Vargr worlds do the same, according to their resources.
And then there are the Solomani supremacists...
In theory, the Ine Givar and the Solomani supremacists are the deadliest of enemies. The IG is a firmly democratic, pansophontist organisation, dedicated to defending the rights of Imperial subjects, regardless of their species or "race". The Solomani groups aren't.
Groups like Rule of Terra and Superioriti frequently engage in terrorist attacks on Vargr and other non-human communities. Frequently, Ine Givar led militias will attempt to stop them. The result is a kind of low-key, sporadic interstellar war.
On the other hand, some IG groups are inclined to conciliate with the Solomani supremacists. This is particularly true of the groups influenced by the Sword Worlds. As I suggested above, they are often operating on (and recruiting from) worlds with a heavily Solomani population, leading (in many cases) to a certain identification with local attitudes. That's not universal, of course. There are also groups that are as scrupulously pansophontist as you could possibly want.
Anyway, I could sit here all day...
The main faction I use in my games is the "Regina Regional Committee of the Ine Givar (Solidariti)".
I will break that down for you. First, because of the factionalisation, there can be more than one Ine Givar leadership body operating in a particular location! In this case, we are dealing with a collection of groups organised around the "Regina Regional Committee".
So, what's a "region"? Well, the idea is that Imperial subsectors aren't always "natural" divisions for military/political purposes. The Regina region is more or less the rimward end of Regina subsector, and various worlds in Jewell, Vilis and Lanth subsectors. Rimward Vilis is dominated by a different faction.
OK, so, what is "Solidariti"? Often, particular factions are identified by aspects of their particular programs, by the names of their publications, or by their most prominent front groups. In the case of Solidariti, it's all three! This is a strenuously pansophontist group, with strong alliances with Vargr groups, and a bitter enmity with the Solomani supremacists. They also orient to mass democratic organisations and movements. So their platform is pansophontist and popular solidarity. The spelling "Solidariti" was adopted as a specific reaction to the human supremacist group "Superioriti", and is meant to symbolise different species standing together in unity, blah blah blah...
RRCIG(S) has an armed wing, but generally prefers to support and participate in local insurrections. As such, they seldom engage in acts of "terrorism", but have been known to make exceptions. Hence the joke: "Q: Where does Baron Delgado-Tukera go for his holidays? A: All over Jenghe." (When claiming responsibility for the Baron's assassination, the RRCIG(S) pointed out that while assassination wasn't their policy, they made an exception in this case.)
Most of their activities are "peaceful", "democratic" activities. Inevitably, this means that their members have a very short life expectancy, and a high probability of spending time in prison, being beaten, tortured or just generally being worked over.
But they have their successes, too... Of course, the actual RRCIG(S) can't lay claim to them, since they are an illegal organisation, and they tend to get attributed to their front groups, or, more often, to their more respectable liberal allies.
Oh, yes, and of course, more traditional "bad guy terrorists" are out there too, including on the turf of the RRCIG(S). "Terrorist hunter" PCs who work for the Imperium are equally likely to be assigned to destroy the RRCIG(S) as they are to fight mad nuclear bombers.
Finally, the Rebellion: most IG factions condemned all of the Rebellion factions. After all, the latter are just different cliques of nobles. A few had mild sympathies with Dulinor, or even Brzk. A more serious matter was the sympathy many IG members felt with the Vargr incursions! And why not - many of them _were_ Vargr!
In fact, that's probably where you would find most IG combatants during the Rebellion, IMTU.
Of course, all of the above is my own take on things. I'm not aware that any of it contradicts canon. The closest it comes is that there is the hint of a connection with Rule of Terra, but that isn't explicitly stated.
Inevitably, some hack will make that explicit, and I will ignore them. This is how heresies are made. I think my version is more interesting.
I will spare you my "Real World"(tm) opinions.
