  Extending earlier results of Godsil and of Dobson and Malnic on Johnson
graphs, we characterise those merged Johnson graphs $J=J(n,k)_I$ which are
Cayley graphs, that is, which are connected and have a group of automorphisms
acting regularly on the vertices. We also characterise the merged Johnson
graphs which are not Cayley graphs but which have a transitive group of
automorphisms with vertex-stabilisers of order $2$. Even though these merged
Johnson graphs are all vertex-transitive, we show that only relatively few of
them are Cayley graphs or have a transitive group of automorphisms with
vertex-stabilisers of order $2$.
