Almost all loops have a trivial automorphism group; almost all groups have a non-trivial automorphism group. What goes on in between?

NB: For this question, everything is finite.

Recently I've been fascinated by the following two observations:

• Almost all loops have a trivial automorphism group (McKay & Wanless, 2005, in the context of Latin squares).
• Almost all groups have a non-trivial automorphism group (in fact, all groups of order $n \geq 3$ admit a non-trivial automorphism).

However, the only difference between loops and groups is "cancellation" vs. "associativity". (Actually, in a loop, an element's left inverse might not equal its right inverse, but the equality of left and right inverses in a group does not need to be included as a group axiom.)

Question: How can we strengthen the loop axioms so as to preserve the property that "almost all X have a trivial automorphism group"?

Question: How can we weaken the group axioms so as to preserve the property that "almost all Y have a non-trivial automorphism group"?

Question: Is the a set of axioms "between" the loop axioms and the group axioms for which "almost all Z have a trivial automorphism group" and "almost all Z have a non-trivial automorphism group" are both false?

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Most semigroups seem to have trivial automorphism (at least that is the result of an enumeration of small semigroups -- see the thesis of Andreas Distler, available on the internet.) A similar question (automorphism groups of free monoids vs. free groups) is at math.stackexchange.com/questions/109600. Note that semigroups and monoids obey the associativite law and yet seem to be non-symmetric in the sense that many have nontrivial automorphism groups. –  Ken W. Smith Jul 14 '12 at 15:55