As @Sebastian-Roth has suggested, by default groups in fog are nothing more than a means to apply settings to multiple hosts at once.

So taking your example, if you apply image a in group a and refresh the page, you will see group a maintains image a as it’s set.

If you then go to group b and it contains all the same hosts as in group a, then apply group b image to image b, it will have set image b to all hosts in group a. So if you go back to group a, it will show image b as being set.

The way these fields are displayed, if all hosts in the group have the same image defined, it will display that image as being selected.

Groups in fog were not designed to be persistent.this is why you can associate a host to multiple groups at the same time.