Generate a set of non repeating pairs of users in multiple groups

I have a dataset that looks like this.

To give some context, there can multiple user groups (odd number of people in it). Each of the groups can contain multiple users in it. So, within each and every group, I needed to select pairs of users in such a fashion that,

A person must not be repeated in any of the pairs, until the entire user list is exhausted. The partial solution below starts pairing users that do not belong to the same group as well. Not sure how to tackle this grouping constraint.

import pandas as pd
import numpy as np

df = [[1, 'a1'], 
      [1, 'b1'], 
      [1, 'c1'], 
      [1, 'd1'], 
      [2, 'x1'], 
      [2, 'y1'], 
      [2, 'z1']]
df = pd.DataFrame(df, columns=['group_id', 'user_id'])
df.head()

I have a partial solution after going through numerous questions and answers.

This solution starts pairing users that do not belong to the same group as well.

Which is not what I want:

from itertools import combinations

# Even Number of users Required
users = df['user_id'].to_list()
n = int(len(users) / 2)

stages = []
for i in range(len(users) - 1):
    t = users[:1] + users[-i:] + users[1:-i] if i else users
    stages.append(list(zip(t[:n], reversed(t[n:]))))
    
print(stages)

Not sure how to store the pairs back into a pandas data frame.

---

Expected Output (which was updated later on):
For group 1 and group 2, note that there can n number of groups:

---

Error while running @mozway’s code:

This error happens for all inputs:

AssertionError                            Traceback (most recent call last)
C:ProgramDataAnaconda3libsite-packagespandascoreinternalsconstruction.py in _finalize_columns_and_data(content, columns, dtype)
    981     try:
--> 982         columns = _validate_or_indexify_columns(contents, columns)
    983     except AssertionError as err:

C:ProgramDataAnaconda3libsite-packagespandascoreinternalsconstruction.py in _validate_or_indexify_columns(content, columns)
   1029             # caller's responsibility to check for this...
-> 1030             raise AssertionError(
   1031                 f"{len(columns)} columns passed, passed data had "

AssertionError: 1 columns passed, passed data had 6 columns

The above exception was the direct cause of the following exception:

ValueError                                Traceback (most recent call last)
~AppDataLocalTempipykernel_14172369883545.py in <module>
     24     return stages
     25 
---> 26 out = (df.groupby(['group_id'], as_index=False)['user_id'].apply(combine).explode('user_id'))
     27 print(out.head())

C:ProgramDataAnaconda3libsite-packagespandascoregroupbygroupby.py in apply(self, func, *args, **kwargs)
   1421         with option_context("mode.chained_assignment", None):
   1422             try:
-> 1423                 result = self._python_apply_general(f, self._selected_obj)
   1424             except TypeError:
   1425                 # gh-20949

C:ProgramDataAnaconda3libsite-packagespandascoregroupbygroupby.py in _python_apply_general(self, f, data, not_indexed_same)
   1467             not_indexed_same = mutated or self.mutated
   1468 
-> 1469         return self._wrap_applied_output(
   1470             data, values, not_indexed_same=not_indexed_same
   1471         )

C:ProgramDataAnaconda3libsite-packagespandascoregroupbygeneric.py in _wrap_applied_output(self, data, values, not_indexed_same)
   1025                 return self.obj._constructor_sliced(values, index=key_index)
   1026             else:
-> 1027                 result = self.obj._constructor(values, columns=[self._selection])
   1028                 self._insert_inaxis_grouper_inplace(result)
   1029                 return result

C:ProgramDataAnaconda3libsite-packagespandascoreframe.py in __init__(self, data, index, columns, dtype, copy)
    719                         # ndarray], Index, Series], Sequence[Any]]"
    720                         columns = ensure_index(columns)  # type: ignore[arg-type]
--> 721                     arrays, columns, index = nested_data_to_arrays(
    722                         # error: Argument 3 to "nested_data_to_arrays" has incompatible
    723                         # type "Optional[Collection[Any]]"; expected "Optional[Index]"

C:ProgramDataAnaconda3libsite-packagespandascoreinternalsconstruction.py in nested_data_to_arrays(data, columns, index, dtype)
    517         columns = ensure_index(data[0]._fields)
    518 
--> 519     arrays, columns = to_arrays(data, columns, dtype=dtype)
    520     columns = ensure_index(columns)
    521 

C:ProgramDataAnaconda3libsite-packagespandascoreinternalsconstruction.py in to_arrays(data, columns, dtype)
    881         arr = _list_to_arrays(data)
    882 
--> 883     content, columns = _finalize_columns_and_data(arr, columns, dtype)
    884     return content, columns
    885 

C:ProgramDataAnaconda3libsite-packagespandascoreinternalsconstruction.py in _finalize_columns_and_data(content, columns, dtype)
    983     except AssertionError as err:
    984         # GH#26429 do not raise user-facing AssertionError
--> 985         raise ValueError(err) from err
    986 
    987     if len(contents) and contents[0].dtype == np.object_:

ValueError: 1 columns passed, passed data had 6 columns```