Auto update of an App after manual installation directly on the phone?

[AC Question]

Auto update of an App after manual installation directly on the phone

Probably someone can me help me with the following constellation:

1. An App shall be pre installed on a phone by copying the apk file to the phone (e.g. via USB) and executing it

2. Updates of this App shall be done with the help of Google Playstore. Means a new version is uploaded to the developer account into the productive environment which shall be automatically installed on all compatible devices (if the devices are correctly configured).

3. In investigations it has been turned out so far that Playstore doesn't update the App automatically as long as it hasn't been installed using Playstore. It seems to be that the Playstore App doesn't know about the already manual installed App on the phone.

4. But if you start the Playstore app, search for the App and open the result it offers to update the App using the "Update" button.

Does someone know if and how the Playstore App connects the own pre installated App on the phone and the uploaded in Google Developer account?
Has someone experience with that?