There are two ways to get the wireless signal into the summer house PC
i) The inbuilt Wi-Fi receiver
ii) A second router, at the summer house end, with sufficient range to send/receive to the main house's router i.e. a "wireless bridge" arrangement
With respect to range, they should be similar (do we expect more oomph from a router than a PC/laptop internal?) but the card may be the speed-limiting factor, if it is old-tech enough and doesn't support newer, faster wireless protocols.
Thus a second router for the summer house ensures best bandwidth.
Footnote: I'm using the word "bandwidth" to mean data transmission speed. Unless I misunderstood, you seem to be using it in the sense of "channels" or "frequency bands". There can be issues where neighbouring routers with only 16 channels start to overlap. Things might be fine at the moment, then goes mammaries up as soon as FIL expands his WiFi footprint and starts overlapping with a more distant neighbour, on the channel his paired routers have negotiated. (Potentially, his and his neighbours' networks all fall over and no-one will be able to work out why!)