13th Btn London Regt or Kensington town hall flag?

[High Wood]

I bought this flag today, the seller had been unable to identify it but I had a pretty good idea what it might be. However, I am not sure about military flag flying protocol so I do not know if individual battalions flew their own flags whilst in barracks or at camp.

If my hunch is correct, this flag should have a connection with the 13th battalion, London Regiment (Kensingtons). If I am wrong it probably flew over Kensington Town Hall. Your thoughts would be welcome.

This flag may be offered for sale.