The following script runs without error in 3.8.5 and raises an error in 3.8.6, 3.9.5 and 3.10.0b1. 

Source:
```
import enum, pickle

class MyInt(int):
    pass
    # work-around: __reduce_ex__ = int.__reduce_ex__

class MyEnum(MyInt, enum.Enum):
    A = 1

pickle.dumps(MyEnum.A)
```

Error (same in 3.8.6, 3.9.5 and 3.10.0b1):
```
Traceback (most recent call last):
  File "/home/thecap/projects/covid-data-model/./enum-pickle.py", line 12, in <module>
    pickle.dumps(MyEnum.A)
  File "/home/thecap/.pyenv/versions/3.10.0b1/lib/python3.10/enum.py", line 83, in _break_on_call_reduce
    raise TypeError('%r cannot be pickled' % self)
TypeError: MyEnum.A cannot be pickled
```

Like https://bugs.python.org/issue41889 this seems to be related to the fix for https://bugs.python.org/issue39587 which changes member_type from int to MyInt. A work-around is in the comment above.

Looking into this I think the root of the problem is the way `reduce` is handled -- currently, Enum's `__reduce_ex__` works by returning the class, and the value to use to lookup the member.  Because that lookup can fail with complex enums, EnumType will sabotage `reduce` if it can't find support in the new enum class.

However, if `__reduce_ex__` working by returning

    `getattr, (self.__class, self._name_)`

then we should be fine, as that should never fail.

New changeset 62f1d2b3d7dda99598d053e10b785c463fdcf591 by Ethan Furman in branch 'main':
bpo-44342: [Enum] changed pickling from by-value to by-name (GH-26658)
https://github.com/python/cpython/commit/62f1d2b3d7dda99598d053e10b785c463fdcf591

New changeset b613132861839b6d05b67138842b579e1af29f9c by Miss Islington (bot) in branch '3.10':
bpo-44342: [Enum] changed pickling from by-value to by-name (GH-26658) (GH-26660)
https://github.com/python/cpython/commit/b613132861839b6d05b67138842b579e1af29f9c

Changing `__reduce_ex__` is too big a change for 3.9, so making the search for pickle support more robust there.

New changeset 3a7cccfd6cd3693e1a2ab65ee05d7f45f8501dfa by Ethan Furman in branch 'main':
bpo-44342: [Enum] fix data type search (GH-26667)
https://github.com/python/cpython/commit/3a7cccfd6cd3693e1a2ab65ee05d7f45f8501dfa

Also fixing the search for the data type so that in

  A -> B -> C

the data type is C and not B.