  Automorphic loops are loops in which all inner mappings are automorphisms.
This variety of loops includes groups and commutative Moufang loops. A
half-isomorphism $f : G \longrightarrow K$ between multiplicative systems $G$
and $K$ is a bijection from $G$ onto $K$ such that $f(ab)\in\{f(a)f(b),
f(b)f(a)\}$ for any $a,b\in G$. A half-isomorphism is trivial when it is either
an isomorphism or an anti-isomorphism. Consider the class of automorphic loops
such that the equation $x\cdot(x\cdot y) = (y\cdot x)\cdot x $ is equivalent to
$x\cdot y = y\cdot x$. Here we show that this class of loops includes
automorphic loops of odd order and uniquely $2$-divisible. Furthermore, we
prove that every half-isomorphism between loops in that class is trivial.
