Hi Tiogiomios84
I think you are mixing up Wi-Fi and WiFi-Direct. You can add your printer to your normal home network, so you won’t have to switch to the HP network every time you want to print something.
There are multiple ways to connect your printer to the Internetbox network. The easiest way is to add the printer using WPS:
Source
The WPS button is the (+) button on your Internetbox.
If this procedure does not work, you might have to change the WiFi encryption from WPA2/WPA3 to WPA2 only.
For this, visit the GUI your Internetbox using a browser: https://internetbox.swisscom.ch
After logging in (If you have never visited this page before, simply set a new password by using the (+) button on the IB).
Enable expert mode and click on Wi-Fi, select WPA2 as Encryption and save your changes.
Best,
r00t