Obviously, the two border face not been extruded as expected. I know boolean modify works for this case but I wish to use extrude.
If i understood the question are you looking for this. No need to remove doubles or delete extra mesh. This addon do the best.
With the Default cube sub divide it twice. With the edge selected scale to Y axis. Select the top 3 Faces and with the Destructive Extrude Addon Push it downwards. Holding control key you can align with the edges.