Hello there, @akilbourn ...
I am sorry to see that your question has been sitting out here unanswered for quite a while without a response.
I think you are confusing a couple different areas of Canvas that can easily get confused because they have similar names. An "Assignment Group" can be set up on your "Assignments" page to group similar assignments together (such as quizzes, essay papers, projects, etc.), and then you can additionally weight those Assignment Groups for grading....such as giving Quizzes 30%, Essays 50%, and Projects 20% of the total grade.
But, there are also Groups you can create via the "People" page of your course. You first create Group Sets, and then you create groups within those Group Sets. So, for example, you might have a Group Set called "Project 1", and then you can create groups of students for that specific project. Also, you could create additional Group Sets for other group projects or assignments...such as "Project 2", "Project 3", etc. And, within those group sets, you could have completely different groups of students than what you had in "Project 1".
So, as it relates to your question about your Canvas assignment...you said you were looking for a group name in the dropdown list of the assignment details. This is not the "Assignment Group", but rather the "Group Set" name. So, I would recommend that you first set up your "Group Set" and the student groups within that group set, and then return to your assignment and set that group set name to the assignment. Canvas will handle the rest.
I hope this information will help to clarify things for you. Sing out if you have any other questions...thanks! Take care, stay safe, and be well.
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