I have been searching for a solution and it should not be that hard. This is what I tried:
I created a cube and rotated it along the Z axis in object mode. In edit mode I switched to the local transform orientation, which looks like this: Which is exactly what I wanted. Next I Extrude, and press X twice to presumably lock to the local X axis. However it ends up pointing in the wrong direction and in the bottom left it says "along normal X" which is not what I want:
How can I get blender to extrude along a local axis? Any help is much appreciated!