D12 Installer does not recognize rtl8xxxu wifi

[Felix Natter <fnatter@gmx.net>]

Dear debian-users,

I put this on a stick and booted it:
debian-live-12.4.0-amd64-gnome.iso

When I boot the live option, the network is immediately loaded and I can
connect to a WIFI network. However, if I use the same stick
(or one with debian 12.4.0 netinst) with the installation option,
the LAN is recognized, but WLAN is not.

If I start a shell from the installer, I can see that the necessary
module rtl8xxxu is loaded (the same one that is loaded in live option).
So I guess it is a firmware issue. How can I get an installer with
non-free firmware (if that is the problem)?
I just found one page that says the (non-free?) FW is always included
in the images (starting from bookworm). Is that true?
Any other idea?

Many Thanks and Best Regards,
Felix
-- 
Felix Natter