  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.
