Hello @hammerin510
In general, I don't recommend trying to make a 5GHz low power mesh interconnect work between buildings – a wired connection will be much more reliable and higher performance.
If you want that outbuilding to be part of the same network created by your primary Google WiFi unit, then you'll want to connect it to the primary Google WiFi unit's LAN Ethernet port. I would connect it to power in your house first and get it added to the network as a wireless mesh and give it a chance to get its firmware updated if needed, etc. Once it's working properly there, I would connect it (still in the house) to the LAN Ethernet port on the primary Google WiFi unit and check the details for it in the Google Home app to make sure it's showing as "wired" connection type. Then, move it to the cottage and connect it through the Ethernet cable run out there and check the connection type again to ensure it's still showing as "wired".
I would also buy a couple of inexpensive unmanaged Ethernet switches to put on each end of that long cable run. That will make more Ethernet ports available to connect wired devices to the "inner" network created by the primary Google WiFi unit in your home (which is a router+firewall), will make more Ethernet ports available to connect wired devices in the cottage (e.g., streaming video devices, gaming consoles, computers, etc.), but most importantly it will provide some electrical isolation in case of a lightning strike in the area. It's much cheaper to replace a couple of $20 Ethernet switches rather than having to replace one or both Google WiFi units.
