Kay of Kingfishers by Constance M. White (Hutchinson, 1954)

I found this in the Green Shed, and bought it a) because the dust jacket was intact and b) because it looked like something I might have enjoyed when I was a child!

To begin with, Kay missed half a term by being ill.  Then, when she did return, it was to find that dear old Miss Benson--surely the nicest headmistress anyone could wish for--had been replaced by young and efficient Miss Oliver.  And Judy, her dearest friend, seemed to have forsaken Kay for a fresh interest--the new Head's Girl Guide Company.

Kay, rebellious and stubborn, turned to Stella Jason, and together they stumbled upon a discovery which involved not only Miss Oliver, but also an archaeological student, the local vicar, and a strange little girl called Bella.

 All right, get your mind out of the gutter.  (Yes, you.  You know who I'm talking about!)  The "discovery" Kay stumbles upon is neither a drug smuggling ring nor a swingers' party, and absolutely nothing naughty is going on between the headmistress, the vicar and the archaeological student.