  This paper exploits adjacencies between the orbits of an ordered set P and a
consequence of the classification of finite simple groups to, in many cases,
exponentially bound the number of automorphisms. Results clearly identify the
structures which currently prevent the proof of such an exponential bound, or
which indeed inflate the number of automorphisms beyond such a bound. This is a
first step towards a possible resolution of the Automorphism Conjecture for
ordered sets.
