How are construction cranes extended to go higher when building skyscrapers to stay above the construction.?

How are construction cranes extended to go higher when building skyscrapers to stay above the construction.? These cranes always stay higher than the construction process but I have never seen how they are extended to stay above all of it. They come in two formats, free standing and those attached to the building. The possibilities that I thought of was that maybe they use another crane to lift the housing and the horisontal arm to put in more modules into the vertical part; or that they extend it by some way by putting modules in at the top or bottom and that the crane has some self-lifting mechanism to accommodate the insertion of more modules. However, I have never seen this being done, nor does the construction of these cranes point to its ability to do so. The only other possibility that I could think of was that the crane is set up at the outset to be higher than the completed height of the building, hence there is no need to extend it but this does not explain how they do it for very high buildings where the crane is attached to the building and seems to grow higher with the building.