SimpleImputer casts `category` into `object` when using "most_frequent" strategy

### Describe the bug

The column `dtype` changes from `category` to `object` when I transform it using `SimpleImputer`.

Here is a list of related Issues and PRs that I found while trying to solve this problem:
#29381 
#18860
#17625 
#17526
#17525

If this is truly a bug, I would like to work on a fix.

### Steps/Code to Reproduce

```python
import pandas as pd
from sklearn.impute import SimpleImputer

df = pd.DataFrame(data=['A', 'B', 'C', 'A', pd.NA], columns=['column_1'], dtype='category')

df.info()

imputer = SimpleImputer(missing_values=pd.NA, strategy="most_frequent").set_output(transform='pandas')

output = imputer.fit_transform(df)

output.info()
```

### Expected Results

This is the output I expected to see on the terminal
```
> > > df.info()
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 5 entries, 0 to 4
Data columns (total 1 columns):
 #   Column    Non-Null Count  Dtype   
---  ------    --------------  -----   
 0   column_1  4 non-null      category
dtypes: category(1)
memory usage: 269.0 bytes

>>> output.info()
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 5 entries, 0 to 4
Data columns (total 1 columns):
 #   Column    Non-Null Count  Dtype 
---  ------    --------------  ----- 
 0   column_1  5 non-null      category
dtypes: object(1)
memory usage: 172.0+ bytes
```

I expected `output` to keep the same `dtype` as the original `pd.DataFrame`.

### Actual Results

The actual results for when `output.info()` is called is:
```
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 5 entries, 0 to 4
Data columns (total 1 columns):
 #   Column    Non-Null Count  Dtype 
---  ------    --------------  ----- 
 0   column_1  5 non-null      object
dtypes: object(1)
memory usage: 172.0+ bytes
```
Observe that the `Dtype` for `column_1` is now object instead of category.

### Versions

```shell
System:
    python: 3.12.3 | packaged by Anaconda, Inc. | (main, May  6 2024, 19:46:43) [GCC 11.2.0]
executable: /home/user/miniconda3/envs/prod/bin/python
   machine: Linux-6.8.0-59-lowlatency-x86_64-with-glibc2.39

Python dependencies:
      sklearn: 1.5.2
          pip: 25.0
   setuptools: 75.8.0
        numpy: 2.1.1
        scipy: 1.14.1
       Cython: None
       pandas: 2.2.3
   matplotlib: 3.9.2
       joblib: 1.4.2
threadpoolctl: 3.5.0

Built with OpenMP: True

threadpoolctl info:
       user_api: blas
   internal_api: openblas
    num_threads: 16
         prefix: libscipy_openblas
       filepath: /home/user/miniconda3/envs/prod/lib/python3.12/site-packages/numpy.libs/libscipy_openblas64_-ff651d7f.so
        version: 0.3.27
threading_layer: pthreads
   architecture: SkylakeX

       user_api: blas
   internal_api: openblas
    num_threads: 16
         prefix: libscipy_openblas
       filepath: /home/user/miniconda3/envs/prod/lib/python3.12/site-packages/scipy.libs/libscipy_openblas-c128ec02.so
        version: 0.3.27.dev
threading_layer: pthreads
   architecture: SkylakeX

       user_api: openmp
   internal_api: openmp
    num_threads: 16
         prefix: libgomp
       filepath: /home/user/miniconda3/envs/prod/lib/python3.12/site-packages/scikit_learn.libs/libgomp-a34b3233.so.1.0.0
        version: None
```

COMMENT:
@rohnsha0 you should wait for the issue to be triaged and a solution to be proposed before you take it. In general, we don't really assign issues to contributors. Also, as a courtesy to the original poster, they should be able to submit a pull request fixing their own issue.

In the future, please don't "take" any issues before proposing a solution. Leave a comment and ask if it's okay for you to work on it, and then start the work.

Labels: Bug, API, Needs Decision, module:impute