You'll start this quest after talking to Sophia Akande, which is... basically the only major task in the previous main story quest, The Puppet Masters. Akande wants a cartographer named Rachel Lockwood to be silenced. Her crime? Knowing too much. It's not a whole lot to go on, and Lockwood has gone into hiding, so she's clearly aware that she stumbled onto something the Board doesn't want her to know. What could a cartographer discover over the course of her duties that would lead to such damning information?
Your first stop in this quest should be to visit the Bureau of Exploration in Byzantium's Paradise Plaza district - just follow the road northwest from the orrery and you'll find the building to the right (northeast). Once in the lobby talk to the receptionist, Sherman Bradshaw, who looks a bit... odd, for a resident of Byzantium. If you pass an [Intelligence] check to out him as the other freelancer Akande hired, and he'll reveal his rather blunt scheme if you call him out. If you're not quite smart enough you can achieve the same end with a [Lie 1] check, then a [Persuade 40] check. That's about the best you can do in this conversation: acquire Lockwood's Office Key to bypass a [Lockpick 60] lock on her door. He'll also refer you to Billingsly's, for what that's worth, and honestly it's not entirely a dud option, but it's not ideal, either.
Key or no, go through a door to the north and talk to Alonzo Vallejos if you want to start the task All Halcyon in a Day. If you want to score some loot, pick the locked door to the northwest [Lockpick 60] (you can hide behind a pillar to avoid Alonzo) where you'll find... a very dead receptionist, a bin and some loose loot.
Anywho, ride the ride the elevator along the northern end of the room Alonzo occupies up to the second floor, then head to the southeastern end of the upper floor to find Lockwood's office [Lockpick 60]. Get inside and search her desk to find Lockwood's Papers, then search her terminal and pick the entries [Messages] and [TO: W. Billingsly] to learn she's hiding in the maintenance tunnels.
Return to the entrance and, before you leave, consider what to do about Bradshaw. You can tell him Lockwood's location, if you want to honor that whole "professional courtesy" thing and he'll offer to sort things out with you down in said tunnels. You can also just outright murder him with no consequence, if you want to ensure he won't show up later on in the quest. Alonzo might run around, but he shouldn't attack, nor should you gain any negative reputation by wasting Bradshaw.
Liberate or Liquidate Lockwood
Leave the Bureau of Exploration building and head down the street a short distance to the southeast until you spot a passage near some stairs to the right (southwest). Go through this passage to find the elevator to the maintenance tunnels, near which you'll find Giles Molina, who will give you the quest Cupid of the Laboratory. There's a good bit of overlap between that quest and this one... mostly because they both take place in the same limited space. Something to keep in mind.
Anywho, descend to the maintenance tunnels and from the elevator, head through a doorway to the west, turn around a corner to the south and continue south to find Lockwood hiding in the southeastern corner of the room. Talk to her and, if you left Bradshaw alive, he'll show up and try to work out a deal with you, splitting the reward 50/50. He gets the kill (and the associated reward) and you get the maps, and your share of the reward. He won't accept it the other way around, but otherwise he's not terrible hard to make a deal with. If you want Lockwood dead without getting your hands dirty, let Bradshaw take care of it and loot her corpse for 2,000 Bits, along with the useless Lockwood's Office Key. If you say you're going to let Lockwood live, or your demand to have the kill for yourself, Bradshaw will attack.
You can also choose to hear Lockwood out, and if Bradshaw is involved, she'll try to bribe you both, but you can only accept if you pass a [Persuade 65] check. Lockwood will pay you 1,250 Bits for your trouble. If Bradshaw isn't around, you can learn from Lockwood that the reason she's being erased is because she was told to erase Edgewater from her maps. Probably doesn't bode well for Edgewater. There's no bribe for letting her go without Bradshaw.
Finally, you can simply shoot her without talking to her, which bypasses the Bradshaw business entirely (assuming he's still alive).
Note: Felix will get annoyed if you kill Lockwood, and demand an explanation. An [Intimidate 25] check is insufficient to appease him, but it's free XP if you can manage it. Passing an [Inspiration 65] or [Determination 65] check will yield better results, and if you can't pass them now, you'll be able to talk to Felix a bit later and try to pass different versions of these checks.
However you go about dealing with Lockwood, return to the HHC Building and talk to Platt, who will transfer you to Akande. If you let Bradshaw killed Lockwood you'll get half the monetary reward, 1,125 Bits and a lesser boost to positive reputation with the Board (around 10%~). If you spared Lockwood you'll gain no reputation and only 825 Bits. If you killed Lockwood on your own you'll gain 2,500 Bits and a larger boost to your reputation with the Board (around 15%). In any event, Akande will decide to move your work relationship forward, asking you to bug Phineas' communication terminal - effectively handing him over to the Board. This starts the quest Long Distance and ends The Demolished Woman. You can demand a larger reward for your upcoming betrayal with a [Persuade 60] check, after which you can engage in small talk with Platt. Neither are terribly important.
You might think that the upcoming quest Long Distance is a point of no return for the main questline, but that's not the case. Not only do you have options for how you resolve it, whatever choice you pick still allows you to side with either Phineas or Akande in the long run, so whether you've decided to work for the Board in the long-term or are still playing both sides, you can continue onto the aforementioned quest without worry.