I found the solution. Thanks!!!
Bug in Group property page
When you are on this page:
And you click on the Group Icon:
And you select the 1st choice: nbb-none
And you click Select
You are taken back to the page:
However there is now nothing to click on if you want to change the Icon from None to something else.
I tried clicking everywhere in that white space, and the icon selection window does not open.
This bug does not exist in the category icon selection page, only seems to be in the group page.
Wow, nice find! It's a really important bug. How I had never tested this before?!