BROTHERHOOD OF STEEL QUEST 04 - Semper Invicta
Triggers after completing Brotherhood of Steel quest 03 - Call to arms
Objective: Return to Cambridge police station
After joining the Brotherhood of Steel at the end of the last quest, you will be tasked with meeting Paladin Danse back at Cambridge police station. Fast-travel back and stride through the front door to see Danse talking to his troops.
Objective: Speak with Paladin Danse
When you walk over and talk to Danse, he will inform the rest of the squad about you joining up (to mixed reactions).
Note: Brotherhood of Steel quest 05 - Shadow Of Steel will trigger after the conversation ends
Objective: Support the brotherhood recon team
You'll now be tasked with helping out Rhys or Haylen with one of their repeatable quests (cleansing the commonwealth or quartermastery). These quests are randomly generated from a table of options, making it extremely unlikely the quest I did would be the same as yours.
For a general guide to both, have a look at the Brotherhood of Steel (repeatable) - Cleansing the Commonwealth and Brotherhood of Steel (repeatable) - Quartermastery in the guide menu.
Some people report only having to complete one of these sidequests to finish this quest, however I had to complete both (having never completed either of them previously) so you may have to do the same. Either way, when you hand back the required number the quest will end rather abruptly.
Note: Talking to Danse afterwards will trigger the Brotherhood of Steel sidequest - The Lost Patrol if you didn't obtain it earlier.