@faacha_lv I am in the same situation with two of my characters. One of them was missing the second key but the quest advanced.
I give the key and nothing happens and the key is gone. I did reset the man hunt but again Kelso took my key and still asking for the same key.

I am waiting to see if we get the reward for the final boss kill I guess.

@ubi-froggard Can you please point me to the forum for the Immortals Fenyx Rising?

Thank you

This bug happened to two of my characters. Both times the leader left the group before finishing the quest.
The only solution is to leave the group before going to the White House and make sure the quest is active before handing in the key.

The only way to fix it for bugged characters is to reset manhunt and redo missions. And never return the key while in a group.
Even everyone else leaves and you are alone, you leave the instance also then hand the key in.

I am going to wait and see how this is going to effect the final mission and the reward before I reset and redo.