The investigator (aka Chief Inspector) can be found outside the museum, or in her lair after she is revealed as the Black Widow. You get into her lair by using the curator's key to get into her house, then clicking on the peeling corner of the painting.
Jump on the light hanging above the Scream to catch the theif and wait for the "investigator" with the police.
The key to the investigator's chest on Counterfeit Island is hidden in the bushes to the right of the cottage. You need to use a stick to reach it and then unlock the investigator's chest to get the next clue.
After you've been arrested, it automatically brings you to the cell in the police station, and then the investigator unlocks you and puts you through a lie detector test.
night right after you get caught the thief is the investigator
it's the little cabin at the very end of the island, if you need help check your map :)
The investigator's house is all the way right (as far as you can go).
The investigator is another of the terms used for the character of the lady Chief Inspector. She is at the museum when you arrive, and arrests you that night. Fortunately, she needs your help.
You don't defeat the investigator -- Black Widow, an art thief, is posing as the investigator to steal The Scream. When you find the painting in the investigator's cottage, she captures you and takes you to her underground lair. (see the related question)
Go to the police office in Counterfeit island. =]
Answer truthfully on the lie detector test.
He is in the clown store.
Go inside the house and peel the painting off the one on the top floor