After the Pinned quest, your first public assignment as the successor to Father is to make a speech. Speak to Father, and he will direct you to a nearby microphone to record an announcement. This is your inaugural speech from The Institute to the rest of the world. After completing this, teleport to Diamond City.
Use the waypoint indicator to find Diamond City Radio. You can reach it easily by passing the alley that includes the entrance to Valentine Detective Agency. Once in the radio station, grab the nearby radio parts in a crate and attach them to the empty slots on the radio console. When you've slotted the parts to the right spots, the quest will update accordingly. Then return to The Institute and make your way to the reactor via the Advanced Systems lab.
Speak to Father, open the reactor door and activate the core. Then start the reactor.
The last step is to meet with the division heads in the conference room, the same room where your promotion was announced during Mankind Redefined. Answer the team's questions however you'd like. Whatever your decisions, the meeting will conclude and so will the quest.