How to remove part of an extruded object in Blender.

Goal:

Subtract the mesh from this protrusion.

Step 1:

Extrude from the surface.

Step 2:

Select the faces of the protrusion:

Step 3:

Hit Shift + D, then hit Esc, then hit 'P'. The purpose is to create an object in the Outliner. This object, which is known as the cutter object, will be used for later step.

Step 4:

Adjust the size by hitting 'S', then hit the key by the coordinate you want to adjust to. In my case, it's 'Y'. You would also need to adjust the size of the cutter object downwards, as to make sure the boolean creates an internal cavity within the main object. How to scale in one direction.

Step 5: 

1) Select the duplicated object (cutter object), 2) add the Solidify modifier on the duplicated object. 3) Adjust the thickness (make sure that it's not too big, or else it will look like Figure 1. 

The purpose of this is because we want to correctly subtract the solid volume. If this step is skipped, the boolean will only subtract the cutter's surface shell as the cutter is not a waterweight solid. 

Refer to Figure 2 below that shows the result of the subtraction with a surface-shell cutter.v

Figure 1

Figure 2

Step 5:

Select the 1) main object, 2) go to modifier, then 3) select the Boolean modifer, then 4) select the newly duplicated object in Object

Step 6:
Hit 'Apply'

Step 7: 

Delete the duplicated object