Some Groups are not available for the Assignment Group field
Upon investigation, it was found that this is happening due to the majority of the Groups in the list having a particular Parent Group.
The Parent Group did not have its Type field populated while the Reference Qualifier was looking for groups with the type of "itil"
The Reference Qualifier function must select groups at all levels of the tree since a group is only selectable if its Parent is selectable.
You will either need to set the Group Type of the Parent to be "itil" or unparent the related Groups.