That is technically doable using this setting as a hack.
Upload the emoji set you want to a website you control, follow the directory structure we use and borrow one of the existing sets name. For example, let’s say you have https://cookieman-emojis.com for this:
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Create a folder at the root of the website named
google -
Upload you custom emoji set inside that folder following the directory/file structure you see at discourse/public/images/emoji/google at main · discourse/discourse · GitHub
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Test if it works by visiting
https://cookieman-emojis.com/google/a.png. Does it shows the correct emoji? -
Go to your own Discourse instance and set
external emoji urltohttps://cookieman-emojis.com. Also setemoji settoGoogle.