So, it's hard to say exactly how your A/V receiver handles zone 2 HDMI output as it is rarely fully described, especially when it comes to HDMI limitations.
I'm not sure exactly how you've hooked up your projector other than the HDZone output. But, if you've used any analog connection (component/composite), you will never see a source as HDMI does not allow video to be converted from HDMI to analog. So, that's not what we will be talking about. We will strictly be talking about HDMI video/audio and analog (red/white RCAs) audio.
HDMI can carry surround sound OR HDMI can carry stereo. It does not carry stereo audio AND surround sound at the same time. So, if you have HDMI hooked up to a TV that is a stereo TV, the audio will never be better than stereo... or maybe the TV just won't handle the audio that is being sent to it.
Think of the projector like a computer monitor. It isn't a TV, it doesn't have audio out, it isn't going to act any other way. In almost all setups, there is a dedicated audio system and a dedicated group of sources that is used with a projector because the projector is stupid. It gets video and can accept stereo audio (maybe) and it has a crappy speaker... Just like most computer monitors. Few have any 'Smart' functionality, they aren't designed to be smart, they are designed to show a really big image.
Stop thinking that you are going to get audio out of the projector or that you should.
Think about the receiver being the point where all audio must (MUST!) go. But, don't think your receiver is all that special either.
Because HDMI can only send surround sound or stereo, your receiver must tell the source what to use. As a general rule, it chooses surround sound. It doesn't include an additional set of mixers inside the receiver to downmix surround sound, or even digital HDMI audio, to stereo, for zone 2. This would add a bit of cost that manufacturers don't want to pay. There are NO downmixing audio extractors which I have seen out there.
So, for zone 2, you need to buy and hookup products which have analog audio outputs. Some Blu-ray disc players still have that functionality. Some will downmix to stereo on a secondary output... But, I'm thinking that almost everything you have has HDMI only. So, you are screwed.
If you put a HDMI audio extractor on ALL of your HDMI only sources, then you will have a stereo audio connection to hook up to your AV receiver to feed zone 2 speaker output.
What you will not have is surround sound. You've lost it.
This is why HDMI sucks, why sources need to be selected VERY carefully, and in the end, you are going to lose something in your setup.
Realistically, you need a second A/V receiver and you may either split sources to both A/V receivers or buy a second set of sources for your receivers. The second A/V receiver can accept the same surround sound as the first, but will downmix the audio to stereo and give you video and stereo audio using HDMI.
Don't complain to me about what a load of BS this all is. I am fully aware.
Some A/V receivers, from what I understand, will downmix to stereo for zone 2 audio output as long as the source for the main zone is currently playing the same thing that zone 2 is playing. This is something to check with your receiver. If you turn on the main zone to your TV, and turn on the same source to your zone 2 speakers, you may hear audio in both locations.
Your second HDMI output is for the HD-Zone feature of your receiver. Make sure that HDMI output 2 of the receiver is setup to be part of the HDZone. This is on page 91 of the manual. Then, follow instructions on page 69 of the manual to select the HDZone and route a source to it. This should get you HDMI video from the receiver to your projector.
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It states that you press the HDZone ON/OFF button on the front of your receiver and you will see an indicator that HDZONE is being controlled. While that light is on, you can switch the HDZone to a different AV source, and you should see that change on the projector.
Now, there is some testing to do, and you may still be able to use an audio extractor because the HDZone may be able to force stereo audio to your sources when it is the selected source. This gets a bit complicated, but go with me here...
HDZone (firestick) HDMI out to the projector. Your projector may force the audio to stereo and you may hear audio. It sucks, because it's the projector's speakers, but you do hear audio from it. The main zone, if you play the same thing, will likely only have stereo audio because that's how HDMI works. But, it will have audio. Zone 2 with the firestick will have nothing because it can't convert HDMI audio at all.
So, this is where you put the HDMI audio extractor in play.
HDZone out into the HDMI audio extractor. It forces whatever source is selected, at that time, to stereo. You then run that audio back into your receiver using red/white analog audio connections. The HDMI audio extractor has an HDMI video output which is connected to your projector. The red/white analog audio outputs go into your A/V receiver hook them to one of the four analog audio stereo inputs. (DVD/SAT/DVR/CD) I would hook it to CD, but it's up to you.
For ZONE 2, select the audio for where you just connected the output of the audio extractor (let's say that is CD). So, HDZone is playing your firestick, but zone 2 is playing CD. CD is the audio from the HDZone! Now you have your audio and you have your video.
When done, turn off the HDZone and audio should return to surround sound for the Firestick.