Fallout 4 created by Bethesda Game Studios.
Images used for educational purposes only.

Prerequisites: Complete Tactical Thinking
Trigger: Speak to Lancer Captain Kells
Reward: 500 XP

Upon finishing up with the Railroad and speaking to Kells you’ll be given this mission, which will send you to speak with Proctor Ingram. She’s still on the Prydwen, back in her original spot by the racks of Power Armor. Your next destination is the Mass Fusion building - and this time, Ingram will be coming with you.

You now have a choice. If you wish to stick with the Brotherhood of Steel, head to the Prydwen’s Flight Deck and get on the Vertibird that’s dangling nearby. This will lock you into the Brotherhood of Steel’s ending (unless you do something to really tee them off, anyway) and make you an enemy of the Institute. You’ll gain the quest Banished from the Institute.

If, however, you decide to report the mission to the Institute (specifically, to Justin Ayo - mind that you need to still be loyal to them in some way for this to happen) you’ll be directed to Allie Filmore, and from there to the Relay, the Institute’s teleporter. Setting foot on the Relay and accepting the consequences will fail the mission, make you an enemy of the Brotherhood of Steel, and give you a different quest instead: Mass Fusion.

Spoils of War is functionally identical to Mass Fusion in most respects, save that you’ll be fighting the Institute as you descend through the Mass Fusion Building, rather than the Brotherhood of Steel. You’ll also have Ingram as your temporary companion rather than Allie. Those differences aside - along with an airborne entry from a Vertibird, which is pretty fun - the strategies and suggestions outlined in the Mass Fusion article apply here, as well. 

Take the Beryllium Agitator back to Ingram at Boston Airport to complete the quest. You’ll gain a new one shortly thereafter: Ad Victoriam