The Amit-Vishne condition for semi-rationality of groups

A finite group is called semi-rational if the distribution induced on it by any word map is a virtual character. Amit and Vishne give a sufficient condition for a group to be semi-rational, and ask whether it is also necessary. We answer this in the negative, by exhibiting two new criteria for semi-rationality, each giving rise to an infinite family of semi-rational groups which do not satisfy the Amit-Vishne condition. On the other hand, we use recent work of Lubotzky to show that for finite simple groups the Amit-Vishne condition is indeed necessary, and we use this to construct the first known example of an infinite family of non-semi-rational groups.