BUG: groupby drops string dtype
- [x] I have checked that this issue has not already been reported.

- [x] I have confirmed this bug exists on the latest version of pandas.

- [x] (optional) I have confirmed this bug exists on the master branch of pandas.

---

#### Code Sample, a copy-pastable example

```python
import pandas as pd

df = pd.DataFrame({"str_col": ["a", "b", "c", "a"], "num_col": [1, 2, 3, 2]})
df["str_col"] = df["str_col"].astype("string")
print(df.dtypes)

avg = df.groupby("str_col", as_index=False).mean()
print(avg.dtypes)
```

#### Problem description

After grouping the string column loses it's string dtype and is object afterwards. This is rather unexpected.
When using a string col as grouping column one would have to manually change the dtypes back to string.
The output for avg.dtypes is:
```python
str_col     object
num_col    float64
dtype: object
```

#### Expected Output

```python
str_col    string
num_col     float64
dtype: object
```

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit           : 7d32926db8f7541c356066dcadabf854487738de
python           : 3.8.5.final.0
python-bits      : 64
OS               : Darwin
OS-release       : 20.3.0
Version          : Darwin Kernel Version 20.3.0: Thu Jan 21 00:07:06 PST 2021; root:xnu-7195.81.3~1/RELEASE_X86_64
machine          : x86_64
processor        : i386
byteorder        : little
LC_ALL           : None
LANG             : en_US.UTF-8
LOCALE           : en_US.UTF-8

pandas           : 1.2.2
numpy            : 1.20.1
pytz             : 2021.1
dateutil         : 2.8.1
pip              : 20.1.1
setuptools       : 47.1.0
Cython           : None
pytest           : None
hypothesis       : None
sphinx           : None
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : None
html5lib         : None
pymysql          : None
psycopg2         : None
jinja2           : None
IPython          : None
pandas_datareader: None
bs4              : None
bottleneck       : None
fsspec           : None
fastparquet      : None
gcsfs            : None
matplotlib       : None
numexpr          : None
odfpy            : None
openpyxl         : None
pandas_gbq       : None
pyarrow          : None
pyxlsb           : None
s3fs             : None
scipy            : None
sqlalchemy       : None
tables           : None
tabulate         : None
xarray           : None
xlrd             : None
xlwt             : None
numba            : None

</details>
