  Kirkman triple systems (KTSs) are among the most popular combinatorial
designs and their existence has been settled a long time ago. Yet, in
comparison with Steiner triple systems, little is known about their
automorphism groups. In particular, there is no known congruence class
representing the orders of a KTS with a number of automorphisms at least close
to the number of points. We fill this gap by proving that whenever $v \equiv
39$ (mod 72), or $v \equiv 4^e48 + 3$ (mod $4^e96$) and $e \geq 0$, there
exists a KTS on $v$ points having at least $v-3$ automorphisms.
  This is only one of the consequences of a careful investigation on the KTSs
with an automorphism group $G$ acting sharply transitively on all but three
points. Our methods are all constructive and yield KTSs which in many cases
inherit some of the automorphisms of $G$, thus increasing the total number of
symmetries.
  To obtain these results it was necessary to introduce new types of difference
families (the doubly disjoint ones) and difference matrices (the splittable
ones) which we believe are interesting by themselves.
