It's impossible.
Autostart is based on authorization tokens that will be provided to the website or mobile application after the user approves the application. For security reasons, these tokens are closely related to the reason they were issued. In particular, tokens and mobile tokens are not interchangeable.
So, you can create your own mobile application to get your own token, but even if you manage to (cookie-) enter it into the browser view, the backend of your site will not be able to use it.
More generally, you are raising a problem, even facebook cannot solve it: say that you are using the facebook mobile application and have registered there. If you open the web version of facebook on the same phone, you will have to log in again. The main reason is the same as above. In particular, any native application cannot set arbitrary cookies in the OS browser. I personally believe that this restriction will not fall, because it will have a big impact on security - imagine how any application can set (and possibly receive) cookies for any website.
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