This takes place in Quest 4: Gorgon. You will follow your mothers clues into the Portrait Gallery and solve the Medusa puzzle to access Mortimer's Secret Room. Inside the secret room you will encounter and chat with Emily and afterwards you will assist her in searching the room. Look at the desk behind her to find an item sitting on it to advance the plot.
You will confront Emily, choose either of the Skill Abilities choices, Call her out on Psychology, She has a Partner, and it's a woman being her sister. She will admit her secret involving her sister and you will unlock this Achievement.