This is less of a solution for the achievement and more of an "in case you cannot get this quest to unlock" guide.
This quest, or any other BoS quests, won't start automatically if you've advanced too far in the main story with other factions. I had already got the "Semer Invicta" achievement and I'd done all the BoS quests up to "Show No Mercy" before going off with other factions. I did a save before just before the warning that proceeding any further would cause you to fall out with the BoS during the Mass Fusion quest. Then after completing all the other faction achievements I went back to that save. I warned the Brotherhood and did Mass Fusion for them, but after that no further quests for the BoS appeared. By then I had been banished by the Institute and after doing some research I realised that I would not be able to undertake the next mission, "From Within: Convince Dr Li to return to the Brotherhood", because she is part of the Institute. I was getting nowhere with Proctor Ingram; she just kept telling me we had work to do. So in the end I randomly went to see Elder Maxson even though there was no prompt to do so or any quest markers; actually I was planning to stick a knife in his head by this point. He said he was annoyed with me because, blah, blah, blah. However, he tells you to take the holotape to Ingram for analysis. This is the tape you obtain by planting a scanner on the Institute network. So you may need to get it back from Sturges or Tinker Tom if you have already given it to them. If they don't give it to you then you will need to pickpocket it from them; it says, "Network Scanner" in their inventory. This then allows you to get the quest to persuade a doctor called Professor Scara, living in Diamond city, to join the Brotherhood instead of Dr Li. This in turn unlocks "Liberty Reprimed" and from there you can then continue on to the the next quest for this achievement: "Blind Betrayal".
I should also point out that I encountered a glitch in the "Liberty Reprimed" which stopped me being able to complete it. Searching online it appears that this is not an uncommon bug if you do things out of order - that is, you don't commit to one faction. After you start the "Liberty Reprimed" mission you're told to follow Proctor Ingram and then talk with her. The dialogue will then have you asking her for the holotape back. After some further dialogue she say she will return it to you, but she doesn't and you will get the same dialogue over and over again. The only way forward is to pickpocket her and steal the "Network Scanner". Make sure you do a save before trying because she will turn hostile if you fail the pickpocket. After that you should be able to complete "Liberty Reprimed" and get the "Blind Betrayal" quest. As there are already plenty of excellent solutions on the actual quest details I won't list them here.
However, for your information, I have copied below a list of all the BoS quests leading up to "Blind Betrayal" in case you'd got so far before going off with another faction and didn't know which missions needed to be done to get back on track.
Reville: A second faction appears.
Fire Support: Join the shootout at the Cambridge Police Station.
Call to Arms: Follow Paladin Danse to ArcJet Systems.
Semper Invicta (achievement): Accept your first quests as an official member of the Brotherhood of Steel.
Shadow of Steel: Hitch a ride on an airship to get a promotion.
Tour of Duty: It's orientation day at the Prydwen.
Show No Mercy: It's time to hunt mutants.
From Within: Convince Dr. Li to return to the Brotherhood.
Outside the Wire: Slip a holotape into a terminal within the Institute.
Liberty Reprimed: Take the Institute down.
Blind Betrayal (achievement): Discover the hidden Cylon.