Begin by enabling the Google Admin SDK in your Google Cloud Console to access the Google Directory API. Create a new project if necessary. Then, set up OAuth 2.0 credentials to authenticate requests. Download the credentials JSON file, which will be used to authorize API requests.

You will need Python for scripting. Install the Google API client library and PostgreSQL adapter for Python using pip:
```bash
pip install google-auth google-auth-oauthlib google-auth-httplib2 google-api-python-client psycopg2-binary
```

Use the credentials JSON file to authenticate your application and access the Google Directory data. Write a Python script to list users or other resources from your Google Directory:

```python
from google.oauth2 import service_account
from googleapiclient.discovery import build

SCOPES = ['https://www.googleapis.com/auth/admin.directory.user.readonly']
SERVICE_ACCOUNT_FILE = 'path/to/credentials.json'
DELEGATED_EMAIL = 'admin@example.com'

credentials = service_account.Credentials.from_service_account_file(
SERVICE_ACCOUNT_FILE, scopes=SCOPES)
delegated_credentials = credentials.with_subject(DELEGATED_EMAIL)

service = build('admin', 'directory_v1', credentials=delegated_credentials)

results = service.users().list(customer='my_customer', maxResults=10, orderBy='email').execute()
users = results.get('users', [])
```

Set up your PostgreSQL database to receive the data. Create a database and a table that matches the structure of the data you want to transfer. For example, create a table for user data:

```sql
CREATE TABLE google_users (
id SERIAL PRIMARY KEY,
primary_email VARCHAR(255),
full_name VARCHAR(255),
is_admin BOOLEAN
);
```

Process the data obtained from the Google Directory to match the schema in your PostgreSQL database. This might involve cleaning data, handling null values, or converting data types:

```python
transformed_users = []
for user in users:
transformed_users.append({
'primary_email': user.get('primaryEmail'),
'full_name': user.get('name', {}).get('fullName'),
'is_admin': user.get('isAdmin', False)
})
```

Establish a connection to your PostgreSQL database using psycopg2 and insert the transformed data:

```python
import psycopg2

conn = psycopg2.connect(
dbname='your_db_name',
user='your_db_user',
password='your_db_password',
host='your_db_host',
port='your_db_port'
)
cur = conn.cursor()

for user in transformed_users:
cur.execute("""
INSERT INTO google_users (primary_email, full_name, is_admin)
VALUES (%s, %s, %s)
""", (user['primary_email'], user['full_name'], user['is_admin']))

conn.commit()
cur.close()
conn.close()
```

Finally, verify that the data has been successfully transferred to your PostgreSQL database. You can do this by querying the table:

```sql
SELECT * FROM google_users;
```

Ensure the data matches your expectations and troubleshoot any discrepancies by reviewing the data transformation and insertion steps.