How to Get Ethernet from a WiFi Extender

Yes, if the extender has an Ethernet port. Depending on its mode, that port may connect a wired client to the extended network or connect the extender to the router for access point operation; check the model's manual.

A compatible extender can bridge a wired device onto the network. The extender itself still needs a working connection to the router, either over WiFi in extender mode or by cable in access point mode.

Only when the extender supports access point mode or another wired-backhaul setup. Connect the cable to the specified Ethernet port and follow the manufacturer's mode and setup instructions.

A cable between the router and access point avoids the wireless backhaul hop, but the result depends on the Ethernet port, cable, router, internet service and client device. A cable from an extender to a client does not make the extender's router link wired.