When I do an inward extrusion the walls of the cube remain, whats going on? I am trying to reduce the thickness of this cube.  When I select the face, extrude along the Z axis, the face of the rectangle goes up, but leaves the walls remaining.  Why is this happening and how can I fix it?  My normals seem to be pointing in the right direction.
I would not use extrude there, instead I would make a loop cut like I shown with blue line, delete the plane I marked with red lines, including the edges, and then fill up the hole using F hotkey.
Let me know if that helped, otherways I'll make a more detailed answer later, when back from work.