Yes @clouise56 very helpful and confirms the fact that this is indeed a phone extension so it won't be connected to your router, and as a result your options are limited.
What I would be inclined to do is to find the other end of it within the house, and from the 'outhouse' tie a (very long) string to the and of if and pull it back through to the house, taking the string with it.
Go online and buy some Ethernet cable along with some wall mounts.
Tie the Ethernet cable (and telephone line if needed) to the string in the house and pull it back through from the 'outhouse'. Mount the wall socket and connect the cables in the 'outhouse' and do the same again in the house.
Using a pre-made Ethernet cable connect one end to your router and plug the other end into the new wall socket in the house.
Down in the 'outhouse' you'll then be able to use it directly by plugging in an Ethernet cable or you could plug in a switch to give you more ports, and or a wireless access point.
Assuming you have power in the 'outhouse' The other and easiest option is to get yourself Powerline adapters, put one in the house and connect it to the router and the other in the 'outhouse' and you'll have the same options as above but without the hassle of pulling cables about.
Of course there are other options available for all of the linked products here, I've only supplied them so you know what I'm talking about.
