This is a complete solution including spoilers.
The following sequence will take you through the necessary steps on Super Power Island :
NOTES : You cannot catch crooks without getting an ID and the handcuffs/files. You cannot get a costume without an ID, but you can get an ID and not a costume. If you turn the water wheel right next to the Sewer Door, it will get too deep, so don't do that. seems to have gotten a bit mixed up, because you can answer the phone without ever talking to the Superhero.