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A router may work depending on which one it is and what settings are available.
Personally I'd buy a switch (you can get a decent one for £20 or so) to plug into the network cable (and the PC) and also plug in a wireless access point (something like this https://www.ui.com/unifi/unifi-ap/ keep an eye out on eBay for cheap ones).
This will keep everything on the same network as your home network.
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I have done this in 3 places in my house, using cheap tp link routers set up in access mode, allows you to use the same WiFi network as in the main house and allows phones etc to easily switch between them, also means you can plug a computer into it. Only thing that gets difficult is anything that needs multicast (only thing I found was you view boxes about 3 years ago.). I bought the cheap routers from Argos, and they work perfectly following this guide
OT and probably a very basic question but I'm technologically challenged. I have a newly built outhouse in the back of my garden that I want to have wifi in. I have already laid a cat6 ethernet cable from the outhouse into the main house that will connect to the router in there (virgin fibre). To get a wifi network at the other end, is the simplest thing going to be to get another router and configure it somehow to broadcast a second wifi network? There will be PC there that I want to connect via a cable too which I presume I can plug into said router?
Would any old router do the job?