Using wifi extender to feed signal to another building via ethernet connection?

Hi,

I have a detached garage, and I'd like to hook up internet access there. The previous owner had ethernet lines installed between the house and the garage. Unfortunately, my internet is coming into the house by comcast cable, and the coax hookup is nowhere near an ethernet jack that could feed the line to the garage. Before installing more wires (either coax or ethernet), I'd like to know if the following might work:

- use a wifi extender with an ethernet port

- place the extender near an existing ethernet jack in the house and connect

Would this work? Would the extender send/receive wifi signal, and then send/receive through the ethernet line to the garage? 

Thanks,

Jeff