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The Friend Activity feature of Spotify doesn’t show music played via Spotify Connect, or directly through Google Home or Google Home Mini; and there is currently not a setting on Spotify to enable this.
I recommend posting an idea for allowing music to show on the Friend Activity feed while using Google Home in the Ideas section of the Spotify Community, here. If enough users vote for your idea, then Spotify may consider implementing it. You can read more about how feedback on Ideas reach Spotify here.
I hope this helps, and let us know if you have any other questions.
Have a good day!
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