For this achievement, you need to fail to solve the mystery, whilst also sending Holly to The Vault.
Since the mystery is solved by
you just need to not do that. No worries though if you accidentally solved one of them. I got the achievement even though I solved the mystery on the 2nd segment.
Speaking of the 2nd segment, this is where most of the work is done. The spirit will
*** Spoiler - click to reveal ***start messing with the SpiritJammer. If you fail to protect the person under attack, they get sent to a room. The order of these rooms doesn't change. First comes The Stage, then The Writer's Room, followed by The Dressing Room and finally The Projector Room.
You will need to sacrifice Holly first, so just keep everyone but Holly safe. By being the first one out, she will go to The Stage and therefore find herself at The Vault in the 3rd segment.
At this point, all you should need to do is play the broadcast to the end.
*** Spoiler - click to reveal ***Who you save or sacrifice shouldn't matter after you've sacrificed Holly, so no worrying about what to do on your way to the 3rd segment.