They were also an organisation almost certainly founded in Sol and probably based there too. We don't know for certain that the orders for the Hades Rebellion didn't originate from Sol before the end of the Great War (possibly even before the start of it since it takes time to build a ship the size of the Hades and it's a fairly good bet that the GTI had some contact with Shivans before the rest of the GTVA military were aware of them).
In other words we don't know how connected the GTI are in Sol. I certainly wouldn't put organising a coup in Sol past them considering how good a job they managed to do of it so far away from the system that is likely their power base.
And who's to say the Hades Rebellion wasn't a direct result of losing contact with Sol, instead? Rampant speculation cuts both ways. However, in this case we're also assuming GTI is still not crazy, and their combat forces would clearly have been out deployed towards or in the fighting. Perhaps there is a great GTI bureaucracy in Sol, but that does not a successful revolt create. This would require a subversion of regular GTA combat forces, something they specifically did not on the exterior and there is no reason they would do on the interior when most of them are probably just trying to find a new job.
Fair point. But it's not hard to stir up resentment at all if you really want to. And we have no idea what civilian politics are like in Sol. It could be that relations between Mars and Earth have always been a powder keg for instance. With the rest of the GTA systems gone, squabbling between those two could quickly result in all kinds of unrest and violence. If 1st fleet intervenes that is very quickly going to make the situation worse.
That's missing my point. First Fleet has seized control of all combat-capable spacecraft in Sol and the means to produce more. You can't
have a shooting war between the colonies now. First Fleet has shielded fightercraft armed with Disruptors to deal with your fleets of Elysiums and Chronoseseses nonlethally if they wish, or Prometheus and Banshees if they don't. The situation is now basically analogous to what would happen if Micronesia's police force was the US Navy. Nobody is resisting because there is no possible way to do so.
Their primary objective would be to maintain that state of affairs to prevent deterioration of Solar politics and the inevitable humanitarian disasters that will come with that. This is a thankless task, probably, but it is imminently doable when you're the only guy with guns.
I've already pointed out that it is easy to generate an external enemy. And we've already seen from the NTF how quickly that can cause an entire fleet to defect.
Which ignores two important facts; Earth is still going to be overwhelmingly represented in the recruiting base, and you can base all your ships in one particular spot thanks to the subspace drive and the best spot is probably Earth from a logistical standpoint. The NTF succeeded in subverting the regional fleet. In this case, there is only one fleet and it's based in only one region. Anyone else up a creek without a paddle.
In fact tying in with the objective to maintain control above, if they really percieve a danger, First Fleet might simply restrict their recruiting efforts to one venue, thereby locking everyone else out, and their basing at another, thereby locking the recruiting venue out while simulatanously frustrating the efforts of the basing venue to subvert the fleet.
Even if I bought those figures, which I don't, you've basically said that an armed rebellion can seize control of Sol (which was my main objection to your post).
Well, sure. I'm looking at this from a purely spaceborne perspective. Anything's possible on the ground, but as far as a shooting war in space goes, it's pretty much out of the question if First Fleet is competent. Even a marginally effective insurgency is not going to possible for at least ten years while you put together some ships to actually conduct it without somehow calling down upon your head the wrath of an Ursa wing.