The 13-28 on my bike is a HG37 like in the Customizing Shimano 7 Speed Freewheels | Harris Cyclery web page and my daughter's bike has one of the no longer available 14-28 HG37s. They both have the lockring with divots to hold the cogs onto the body. Webpage and "All the cogs are splined and held on with a threaded lockring" credit to Darth Lefty. Since the 14-28 has 14,16,18,20 on the 1st step I should be able to start with a 16 and make 16,17,18,19 on the 1st step using the 17 and 19 from the 13-28 then file on the 21 from the 13-28 so the splines fit the 2nd step of the body and finish off with 21,24,28. The finished freewheel would have 16,17,18,19,21,24,28 with percentage jumps of 6.3, 5.9, 5.6, 10.5, 14.3 and 16.7%. I can use chainwheels of 50 or 52 then 39 or 40 and a small 28 for uphill.

I'll replace my daughter's 14-28 with a 13 - 28 or a 14-34 mega range since she has a long cage Tourney RD.

The alternative would be to get a Shimano cassette hub in 7 or 8 with an 8 sp downtube shifter for my bike and use my 7 sp shifter on my son's bike (Trek 1000).

This is the most knowlegable forum I've ever come across.