Telegram beta is a standalone app that receives more frequent releases with features that are not yet available on the Stable version. If you already have a beta app installed, updating process should be working within the app itself.
If you already know how it works, this question will sound obvious to you but unfortunately, it is not always that straightforward.
Normally, when the Telegram team releases a new version, the next time when you open the Telegram beta app, you will see a pop-up window saying that a new version is available. This pop-up allows you to start downloading process and as soon as it is finished you will be prompted to instal a new APK for Telegram beta.
In most cases, it should work but sometimes it also happens that the update is getting stuck or just disappears. These issues could be related to the OTA update process bugs specific to certain Android device models. In case if you faced this issue, you can side-load the latest APK directly from the Telegram App Center page.
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