The only option that comes to my mind requires your app to be opened by the InAppBrowser
in a separate window where you can execute
additional script on an event fired. Even then you will have to do some changes on your web app.
Here is a blog post
providing detailed description how you can obtain the needed data. Then you will have to store it in your mobile app local storage and use it to log the user next time he wanta to access the web app.
I hope this would do the trick on your side.
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