You can use a basic IR repeater system, like those available from Xantech and others. You put an IR receiver under the screen and run a wire back to the equipment rack in another room. Locally the projector is powered on and off directly through the remote, and the remote will use the IR repeater to send IR signals to the devices in your rack. This is decades old and works fairly well.
You can also use the above mentioned RF repeater systems if your remote supports them. Some remotes use both IR and RF, so your projector commands would be sent out locally to turn it on and off, but the IR commands that need to go to the other room would be sent via RF to a receiver, which would then break them out to IR at the rack.
I personally use an entirely RF remote system which broadcasts all my commands via RF to my controller and the controller (in the rack) then converts them to IR, serial, relay, and IP commands as appropriate to control and run my A/V system. It's more expensive than the more baseline IR/RF Harmony Remotes from Logitech which you can setup yourself.
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