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?