I see that people did not really understood your question.
This solution might work.
In a nutshell, is to add a new protocol to get the authentication token from the server to the app client.
PG App calls a web method registering a request to Auth:
POST BeginGethAuthToken id = pickup Guid pswd = pickup Password
BeginGetAuthToken can be a web method in your login page.
Server associates the pickup guid with the pickup password
PG Apps start the authentication flow by opening the log-in page using the InAppBroser
mywebapp.com/AppLogin.aspx?id={pickup guid}
The pickup guid is set as a cookie for your webapp domain.
User completes the authentication using the InAppBrowser.
AppLogin.aspx on successful authentication associates the pickup guid with the authentication tokens and ask the user to close the browser to go back to the app
the PG App picks up the authentication token by providing the pickup password.
note that for security the server needs to add expiration policies of pickup guids such that only one app can use a guid at a time and restart the process in case of collisions.