TST(string dtype): Resolve xfail in groupby.test_size
- [ ] closes #xxxx (Replace xxxx with the GitHub issue number)
- [ ] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [ ] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [ ] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [ ] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

groupby does inference on the group labels across the board.

```python
df = pd.DataFrame({"a": [1, 1, 2], "b": [3, 4, 5]}, dtype="object")
gb = df.groupby("a")
result = gb.sum()
print(result.index.dtype)
# int64
```

While I agree long-term I'd prefer to preserve object dytpe, I do not think we should be changing this at this point.
Owee, I'm MrMeeseeks, Look at me.

There seem to be a conflict, please backport manually. Here are approximate instructions:

1. Checkout backport branch and update it.

```
git checkout 2.3.x
git pull
```

2. Cherry pick the first parent branch of the this PR on top of the older branch:
```
git cherry-pick -x -m1 354b61f88bc0523d4bb9f3cfe1d6c12f9a3d6567
```

3. You will likely have some merge/cherry-pick conflict here, fix them and commit:

```
git commit -am 'Backport PR #60711: TST(string dtype): Resolve xfail in groupby.test_size'
```

4. Push to a named branch:

```
git push YOURFORK 2.3.x:auto-backport-of-pr-60711-on-2.3.x
```

5. Create a PR against branch 2.3.x, I would have named this PR:

> "Backport PR #60711 on branch 2.3.x (TST(string dtype): Resolve xfail in groupby.test_size)"

And apply the correct labels and milestones.

Congratulations — you did some good work! Hopefully your backport PR will be tested by the continuous integration and merged soon!

Remember to remove the `Still Needs Manual Backport` label once the PR gets merged.

If these instructions are inaccurate, feel free to [suggest an improvement](https://github.com/MeeseeksBox/MeeseeksDev).
                 Manual backport in https://github.com/pandas-dev/pandas/pull/60782