Way to unlock the mission: The opportunity to start completing this mission appears after you complete the Deactivate the Jamming Signal mission, i.e. after you sabotage Enigma's first communications tower, located in the Coventry district.
You need to go to Enigma's headquarters located in the South-Eastern part of Arkham, in the Burnley district (screenshot 1). After you get there, ignore the goons occupying the lowest level and locate the catwalk shown in screenshot 2. Initiating the Detective vision here will allow you to locate the weaker fragment of the wall. Spread the Explosive Gel and detonate it.
You can now enter the Enigma's HQ. Use the elevator and make it over to the room at the end of the corridor. The attempt do destroy Enigma's computer will fail but, you will learn instead what you need to do to stop the madman. The plan is that you destroy the Network Relays all around Arkham, interrogate Enigma Data Handlers and obtain Enigma Datapacks. Since you cannot do anything, as for now, leave the HQ.
Note- detailed information concerning all of the abovementioned actions can be found in the following chapters: Comms Towers, Network Relays, Enigma Data Handlers and Enigma Datapacks.
Return to Enigma's HQ after you have neutralized all of the 70 Network Relays. You can now destroy the central console to sabotage the madman's main server, for which you will be rewarded with 50000 experience points. This is, of course, not all, because you can still interrogate the Enigma Data Handlers and find Enigma Datapacks.
[New character trophy: Enigma]
[New concept art: Royal - Cards Corridor]
Visit Enigma's HQ for the last time after you have found all two hundred Datapacks, i.e. after you have collected all twenty Extortion Files. Again, go to the area with the main computer, but use the door on the left this time. Use the cryptographic sequencer and determine the password (QUESTIONSANSWERS; the screenshot).
Jump over to the neighboring ledge and examine the wall with the question mark painted on it. Ram through the wall. Your character will find here the first Riddler Trophy, the collection of which completes this mission. The main rewards for the completion of the Enigma side mission are 100 thousand experience points and a new outfit (you can try it on after you return to the Batcave).