Moufang permutations over small abelian groups

Moufang permutations are certain permutations on an abelian group $X$ that differ from an automorphism of $X$ by a symmetric alternating biadditive mapping. It is known that every finite split abelian-by-cyclic $3$-divisible Moufang loop is obtained from a Moufang permutation of the abelian normal subgroup. In this paper we investigate Moufang permutations for small abelian groups. We prove that a finite abelian group $X$ possesses non-automorphic Moufang permutations if and only if the $2$-primary component of $X$ is of order more than four and is not cyclic. The automorphism group of $X$ acts by conjugation on the set of Moufang permutations of $X$ and the orbits of this action provide a partial answer to the corresponding isomorphism problem. We explicitly find all Moufang permutations for small abelian groups, including small elementary abelian $2$-groups.