From the Android documentation ( http://developer.android.com/reference/android/webkit/WebView.htm l )
Cookie and window management
For obvious security reasons, your application has its own cache, cookie storage, etc. - It does not use browser application data. Cookies are managed in a separate thread, so operations such as building an index do not block the user interface stream. Follow the instructions in CookiesyncManager if you want to use cookies in your application.
So, the short answer is no, you cannot access anything that the main browser stores in your WebView application.
I'm not quite sure what you can do to get around this. Can't you just save the session data on your server in the database and associate it with the user account. Then you can have an application for storing their data for login and automatic login, which means that they will not need to enter a username / password each time they use the application, and it can simply synchronize their session data every time . To the user, it should look pretty smooth.
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