You have to talk to Katy, and she will ask you to tell the children a story about the Commonwealth. You will need to wait for the next class to start (around 9AM) unless there is one already in session. When the class starts, you have to talk to Katy again and confirm that you're ready (if asked, Katy will say to stay away from overly-violent topics, but it does not seem to change the reward you receive).
Instantly, one of the children will ask you if you've ever fought a deathclaw. From here, you can tell the straight truth about the deathclaw in Concord, say you fought it off with three shots from a pistol, or redirect the conversation elsewhere.
At the end of the story, the children will thank you and you shall receive a Grognak the Barbarian comic.
If you have started Hole in the Wall, it is not possible to progress this quest as Katy will stay in her room at all hours and class will not begin at its normal time. Complete that quest first in order to make this one continue as normal.
The story recounted to the children is the events of When Freedom Calls. If you enter Vault 81 prior to traveling to Concord and saving Preston and the settlers, Katy will not initiate this quest.
Even if Preston is the active companion during the retelling of the story, he has neither a reaction to the Sole Survivor's account of the events nor does his affinity change.
pcps4 This quest will become stuck if you have failed the Hole in the Wall quest. [verification overdue]
On PC, it can be forced to progress by using the console command setstage 1338b9 200.
pc This quest will sometimes be stuck if you try to talk to Katy while she is sitting down. Type resetquest 1338b9 followed by setstage 1338b9 100 and you should be able to proceed with the quest. You can also be locked out of the quest, if Katy wants to talk to you by herself to start the quest, but walks away during the conversation. The resetquest / setstage fix mentioned before works perfectly. [verified]
pc The first objective to this quest is bugged and will not be marked as complete, even if the quest is successfully finished. This makes the second objective impossible to obtain without console commands. [verified]
On PC, this can be remedied by opening the console after agreeing to talk to the kids and typing setstage 1338b9 200 to complete the first objective.
pc When finishing the quest, it will sometimes pause dialog when switching from Katy telling the kids to say "Thank you" and when the kids will say thank you. To fix this, move the mouse until you are not in the conversation and rejoin. It might not work at first. [verified]
ps4 After finishing the quest and waiting nearby for a time without a companion, Preston may appear in the classroom. He will react as normal, and begin to leave the vault to wherever he is supposed to be/assigned to. [verification overdue]