If a ship engages (or otherwise fires) at init 7 and kills an init 6 ship, then the init 6 ship will be removed before the initiative 6 band starts. This is because we were in the init 7 band when the ship was destroyed, and destroyed ships are removed at the end of the current initiative band (giving ships of the same init the chance to fire back, but removing anyone else). Init 7 works like all the other initiative values.
So the init 6 ship will never get to engage -- it's been initiative killed. Top-init aces aren't terribly used to that, but sometimes it's their turn. :D