I suggest a general algebraic approach for this problem, since to me the major thing that is changing is the idea of 'many". Although you haven't said so, you seem to be asking about a demarcation in the lattice of varieties of algebra with one binary operation where one side has algebras with more than one automorphism versus athose those that hav e have only one. The latter are called rigid, and since no nontrivial variety has only rigid algebras (think of powers), you will need to have a good technical definition of 'many'. Perhaps pseudovarieties are the classes of interest.
I recommend looking at studies of rigid algebras. If you are interested in equational formulations which promote rigidity, you could do worse than looking at versions of primal algebras, which are very rigid. The varieties they generate are called arithmetical, and therein might liie lie part of the answer you seek.
Gerhard "Ask Me About System Design" Paseman, 2012.07.14
