Okay so you guys thought the syndicate was bad right? Imagine taking on the world! Wait.. maybe that's too big... Or is it?! /Dramatization
One day in Steelport as the boss sits on his chair watching the News he notices... Something.. <--- You can fill that gap in there
And he comes up with an idea and says, hey buggas(in english voice), let's do something nice.
What was that nice thing he was talking about? Well with the bosses' humour it would probably be something down the lines of promoting their products or accomplishing big humanitarian projects with their money and getting the spotlight on TV being seen as a real "Saint".
That would increase the revenue of their franchise, hence become more influential.
Etc etc etc (I will fill in this gap later).
[Small Details] The Saints find out Pierce has another name in a place like Jamaica or something.. There's a meeting in the white house where a person is showing graphs that say people rather vote for the boss of Saints(or someone else in the Saints) for president and they aren't even running for president, however, it catches the Saint's attention and perhaps some.. Other people.. Dun dun dun...
Face it, the Saints are bigger now, why would you dim them down again like in SR3? Unless it makes perfect sense or is hilarious :D
In a more sophisticated war, the Saints will have to get more technological and have a moonbase, satellites(thanks Kinzie), rockets, and more.
Now now.. This is starting to remind me of Dr. Evil when he had all that stuff and blackmailed pretty much every country leader for money.. But the boss ain't like that!
So, you might have something like Star Wars going on with the ships shooting eachother, trying to blow up the deathstar(if that happened, there could be a cutscene when Shaundi or Pierce is flying in the ship yelling out: "I swear I seen somewhere, it's so familiar! **** it's like on the tip of my tongue!" Boss: "Really? You seen this big-ass piece of-" Shaundi yells: "ROCKETS!!"
Okay I'm tired here I'll edit this later @.@