You can get the groups of a user with request.user.groups.all(), which will return a QuerySet. And then you can turn that object into a list if you want.
for g in request.user.groups.all():
l.append(g.name)
or with recent Django
l = request.user.groups.values_list('name',flat = True) # QuerySet Object
l_as_list = list(l) # QuerySet to `list`