This topic came up again as an offshoot of a Mabie Todd post. I figured I should share my answer and pictures with the repair Q & A crowd. Here is my inner cap puller set up. From left is the tap, and the device for pulling. On the right is the extra thrust washer (it was a set of two on line), and the plain washers that I used before upgrading. I now have a similar set up in three in different sizes.
It costs a little, but there is a lot of satisfaction in doing it yourself. It should run less then $25, and then you can replicate for other cap sizes for only a few dollars more for each additional size.
Go to Harbor Freight and get a cheap set of taps and dies that look like they span the size of pen caps. I got this one, and it is only $16.49. Don't forget your 20% off coupon and you cost is $13.19 plus tax.
Once you have it, see which tap goes inside the inner cap, and which is the first one that is too big. Use the one that is one size too big. Then use it to cut threads into the inner cap. This will not affect the function later, and appearance does not matter.
Next is a trip to Home Depot or Lowes. You need to buy a bolt to match that tap - the tap will tell you diameter and thread count, and the tap and die set you buy will have standard sizes. You also want to buy a wing nut to match the bolt.
The best last piece is ebay special order. If you want to do it easier, while you are at home depot get two washers with a hole as close to the diameter of the bolt as possible, and large enough to be larger than the diameter of the cap.
Special order is to get a "thrust washer" on ebay meeting the same criteria - inner hole close to the size of the bolt, and then wide enough to be larger diameter than the cap. These usually come with the flat washers you will also need. They look like this, and often cost under $3.00 from China
Put the wing nut on the bolt so the flat side is facing the threaded end of the bolt and twist it way down the bolt. The add the washers or the thrust washer assembly (thrust washer between two flat washers. And thread the bolt into the inner cap.
I never had a problem with two normal washers, but there is potential for friction at the cap lip with this set up. The thrust washers were cheap enough, and then there is never any rotation of the washer in contact with the cap lip. You always get pure pulling action and nothing else. I usually use a vise grip to hold the bolt. Be sure to grip it by the head or next to the head so you don't wreck parts of the threads you will need to use later. When it is particularly difficult, I use pliers to help turn the wing nut.
Now start twisting the wing nut until everything is snug to the cap lip. And then keep on going. It will pull out the cap. Be sure that you have soaked the cap first, or preferably run it thru an ultrasonic jewelry cleaner to get out the crud.