In Star Trek Into Darkness, precisely what does Harrison
say about the reason why his people were frozen and when were they supposed to be awoken, and is the information he gives correct?
The quote from the film is
Khan: A remnant of a time long past. Genetically engineered to be superior so as to lead others to peace in a world at war. But we were condemned as criminals, forced into exile. For centuries we slept, hoping when we awoke things would be different. But as a result of the destruction of Vulcan, your Starfleet began to aggressively search distant quadrants of space. My ship was found adrift. I alone was revived.
Which ties up very nicely with what we know of Khan from the original series episode Space Seed.
SPOCK: There is that possibility, Captain. His age would be correct. In 1993, a group of these young supermen did seize power simultaneously in over forty nations.
KIRK: Well, they were hardly supermen. They were aggressive, arrogant. They began to battle among themselves.
SPOCK: Because the scientists overlooked one fact. Superior ability breeds superior ambition.
KIRK: Interesting, if true. They created a group of Alexanders, Napoleons.
SPOCK: I have collected some names and made some counts. By my estimate, there were some eighty or ninety of these young supermen unaccounted for when they were finally defeated.
Obviously the key difference in timelines is that in the original series, Kirk and his crew were the first to discover the Botany Bay whereas in the Abramverse timeline, his ship was discovered some ten years before as a result of Section 31's "aggressive search" for technologies that could be used to defend the Earth.
The implication is that only he is thawed by [redacted] and the others are used as leverage to assure his compliance - though there's some question of whether he's being honest about how eager he was to be helpful anyway.
The answer to Lister's "why didn't he do it himself" is that he was in the process of trying to do so and was stopped, resulting in the containment within the cough high speed transport pods cough. Since it's established that the technology to do the thaw isn't easily accessible/understood by the contemporary sorts it seems likely that [redacted] couldn't just thaw them himself, he'd need to get them off somewhere to do it when/where he had more time & resources.