You should make a temporary user account just for troubleshooting this issue, you can delete the account and profile folders once you know how IE11 and Google interact.
Make that temp account an admin. We might need it to delete some stuff from your normal Windows user account profile folder.
Callendar, I followed your advice and uninstalled IE 11, rolling back to IE 10. That seemed to work although the system hung at "Configuring Windows Updates" and I had to hard reboot in safe mode to complete. IE 10 then remembered my Google passwords. I tried indtalling IE 11 x64 with the correct installer and with Windows Updates but the update failed repeatedly with the error mentioned here:
WindowsUpdate_00009C59" "WindowsUpdate_dt000" - Windows 7 - Microsoft Community
I ran the Windows Update Troubleshooter and retried a few times, then I then did a system restore back to just before uninstalling IE 11 and everything was back to normal in my system including having a functioning IE 11 install. And the best part is, now IE 11 remembers my google login credentials opens my Google Calendar page signed in to my Google account.
This is voodoo but I'm happy to report the issue appears to be fixed. Will update if the fix holds or not.
When system hangs at "configuring updates" it's always best to see if it will complete - even if it takes a few hours. Anyway the uninstall if IE11 probably fixed something (possibly in registry). I agree that if it's now working then leave it as it is.
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