@kenji Thank you for bringing this topic to the Sonos Community. We do not have any supported way to have you get the projector TV audio to Sonos. As in your previous post to the Community, maybe users are able to step in with ways to make this work for you and your setup.

As I have yet to come across a projector that has HDMI-ARC, or handles audio in any way that would be helpful to Sonos, I think your best bet would be to go with an HDMI audio extractor, that will output audio over an optical output. That can then be fed to the Beam using the optical > HDMI convertor supplied with it. I use an HDMI switch/audio extractor with a Projector and Playbar, with a cable box and Chromecast on the input side of things.

The HDMI audio extractor can take input directly from a computer, or from a Chromecast or Apple TV. To get the best experience, you need to make sure that you are outputting DD 5.1 to the Beam, otherwise you will get Stereo or silence, but no surround.

I wouldn't recommend casting your entire screen directly to a Chromecast, as it is very dependent on direct connection between the Chromecast and any device you are streaming from. I have my video on a NAS in my network, and stream it to my Chromecast using BubbleUPNP, butthere are other solutions. Having the Chromecast do its own streaming directly from the internet or the local network, and using the phone or other devices as a control only, greatly improves reliability of the stream and battery life on your devices.

The downside to using optical as a go-between is its limitations, should Sonos decide to add other codecs to the A/V products in their S2 ecosystem.