Whichever way you chose to finish Picking Up the Trail, you will enter the vault in the same spot. Proceed through the vault until you get to the Test Labs to find an intelligent super mutant named Fawkes. You can either set off the fire alarm to free Fawkes who will help you fight and retreive the G.E.C.K. from the highly irradiated section of the vault, or attemt to brave the radiation and get the G.E.C.K. yourself. Either way, once you attempt to leave the vault with the G.E.C.K., you will be knocked out, ending the quest.
See Finding the Garden of Eden for more specific info.