The Government/FBI
Well, for the many, many plot holes (in a very good TV show that I really enjoyed when I was younger), the FBI makes no sense.
Are they stupid, how the heck did they become agents?
First off, they admit that they may have something, "proof of alien contact" as Ms. Topolsky said in the Liz diary, when it goes missing. Then for some reason, the super s3cret alien division of the FBI gives up trying to obtain it, just because Liz found it again? I am sorry, but I just can't swallow that.
If it was that interesting to them before, it should still be interesting.
They want Max's blood at the hospital, and they prove it was Alec's. OK, so why don't they do the same thing they did to Alec to get the blood from Max? Even if they find it doesn't work, they can't orchestrate anything, like a bully who just picked a fight? A kid would be able to think of something to get his blood easily enough, why can't they? Are they really that inept?
Also, why don't they just catch one and test them? "They need proof" is BS. So get proof, but for some reason the government expects to sit and wait for something to happen.
ANY SMART INVESTIGATION like this would try to be manipulating events at the very least, even if they were passive about it. To force the truth to come out. These people seem to sit around and do nothing but spy on these kids (and they even get an F on that).
Another thing, it is clear from Topolsky that they are willing to kill for the mission, being brutal, dismemberment to prove a point (even to a Federal Agent), coverups that even the president is need to know. They kill people without oversight, that is even what Topolsky said.
So, if that is true, why does it even matter that they need proof? These people are brutal thugs on a mission. If they do happen to all be human and it is a hoax, they could just kill them if need be, after finding them not aliens. It wouldn't be the first time, and they are breaking their own patterns doing this BS.
Don't get me started on the amateur spying. They can't get a single conversation recorded from them, when they often talk so publicly?
I am willing to forgive the agent pierce turncoat suddenly destroying the division, on account that we don't know what weird government agreements they really operate under on a president-need-to-know division. But it still reeks of poor writing.
After division is gone, at the very end when they come into play again, it becomes clear that they had the ability to get evidence quickly all along (the same agents, who couldn't before) after the general gives them the go ahead. The pictures, spying, manipulating events (shooting, debriefing the woman) of the sort they should have done before. It proves it wasn't impossible after all? Or did they just suddenly become better all the sudden?
Also at the end, why didn't they continue to spy on them to get intel, possibly for the benefit of technologies, contact, or what? Even to know why they are here?
Instead, the general just orders their execution. This is also forgivable, and possibly understandable as a real possible outcome. But also unlikely.
If our government employed people like this, I feel sorry for our national security, and national interest. And just for humanity in general to think people that stupid would be put in such powerful positions.