Microsoft HomeGroup does work after all. After days of trying off and on to get HomeGroup to work with my home network computers, I discovered that the problem was my router. It was not the computers or the software. My CenturyLink ZyTEL PK5001Z DSL modem and router was not configured for the IPv6 Internet protocol. Apparently HomeGroup requires that its network use IPv6, rather than the old IPv4 protocol.
Instructions on how to enable IPv6 on the CenturyLink PK5001Z modem can be found here. http://internethelp.centurylink.com/internethelp/modem-pk5001z-ipv6rd.html
After I enabled the IPv6 protocol on the router, rebooted the computers and waited for a while, the option to create a HomeGroup suddenly appeared on a laptop computer. The option to create a HomeGroup had not appeared since the major Windows 10 update. The only option had been to join an existing HomeGroup, and after a long pause, Windows would say that the HomeGroup it was trying to join no longer existed, but it still would not display the option to create a new HomeGroup.
The tricky part of this for me was that the computers all showed that they were configured to use IPv6. Trying to fix HomeGroup, I would often get error messages saying a protocol was missing, but there was no information about what it was or how to install it. There are apparently others with similar HomeGroup problems, and I found solutions posted on the Internet, but none worked for me. I wonder what I have done to normal networking functions by trying some of the suggested solutions, deleing files and changing registry entries, but so far HomeGroup and the other network functions seem to be working correctly after fixing the router.