The Copenhagen public transportation system had been confusing me for a long time.

I usually travel between Aarhus and Copenhagen. The train ticket to any point around Copenhagen is the same price. The airport, Copenhagen H, and even Helsingor are all the same. So, can I travel to other places with the same ticket free of charge? For example I just buy a ticket to Copenhageb H but continuing to the airport? I know that in Aarhus if you get a train ticket, you can travel from any place in zone 4 on the same day. Is that the same for Copenhagen?