This (and in fact the village of Brimstage) was known in Tranmere as the "Three Mile Limit", as it was three miles from Tranmere and this entitled you to be called a traveller and to demand food and drink when the Inn was closed.
You are correct. It is the old school house at Brimstage. Here's another..
Description English: The old school house, Green Bank, Brimstage, Wirral, Merseyside, England.
Date 9 June 2013, 11:23:40 Source Own work Author Rept0n1x
Humankind has not woven the web of life. We are but one thread within it. Whatever we do to the web, we do to ourselves. All things are bound together. All things connect. ~Chief Seattle
The original inn of that name [Red Cat] stood in the village of Brimstage until it was demolished to make room for the village hall erected in 1932 by Lord Leverhulme. The village is overlooked by Brimstage Hall and its chantry chapel. One of the corbels of the chapel shows a "cat" carved in the red sandstone. The family of Lady Margery de Hulse, who built the chapel in the 14th century, was a branch of the Barons of Montalt, whose coat of arms was a red lion rampant. It is thought that the mason who worked on the chapel had never seen a lion, and, on being informed that it was a member of the cat family, he carved a cat's head with a snarl, or, as it appears, a "grinning Cheshire cat!"
Extract from "The Story of Greasby" by John Williams, 1978
Taken from:http://www.greasby.btck.co.uk/pubs
Humankind has not woven the web of life. We are but one thread within it. Whatever we do to the web, we do to ourselves. All things are bound together. All things connect. ~Chief Seattle