Only available if your Commander Shepard has the Colonist background when you create the character.
You can resolve the assignment and save her life by having enough talent points in Intimidation or Charm.
You can also resolve the assignment by asking Lieutenant Girard to kill her with the sniper watching the event.
You can also cause her to kill herself by attempting to get too close to Talitha.
You can also walk away from the mission entirely by boarding the Normandy and leaving the Citadel, the mission will resolve with the Lieutenant solving it himself.