Update an app that is not in Google Play I have an app which is currently in beta testing. With every new version I have to send it to the users, they have to download it, uninstall the previous version (sometimes), and then install the new one.

Is there a way to update an app without it being in Google Play?
You might provide them a link! A link on your website, that would provide them with the .apk file. After downloading, the user would install that file in his android phone. 

After installing, the previous version is auto-deleted and the new version is installed! User data is saved. But I have never used the developer tools from Google Play. So I am not sure, whether the Google would mark it as spam or not, because some user try out Verify And Install instead of Package Installer. Verify and Install is a Google service to check whether the app is spam or not. 

So I think, if the user tries Verify and Install instead of the built-in installer, he might get some errors, saying this signature is not found on Google Play! Or something like that.

But just my guess, if your beta is placed on Google Play, then they won't show up any errors, as they would find a signature on Google Play too. 

Hope I am not wrong: Because I have never used either Verify and Install or Google Play Developer tools. So I might be wrong somewhere. But still there is a functionality to install apps, but that also depends whether the user is allowing the installation of Unknown Apps (Apps which are not downloaded from Google Play).