Manchesters, it is indeed a picture of the early FANYs. The lady with the dog is Grace Ashley Smith, later MacDougall - a strong driving force of the early FANYs. The dog was their mascot, named Caspar. The picture is documented as being taken in 1910. As far as I know, the other ladies are not identified.
Here's a slightly later picture that shows a group of FANYs at one of their yearly camps in pre-war 1914. They are also wearing the round Red Cross patch.
Terry