  In 1911, Burnside asked whether or not there exist groups that have an outer
automorphism which preserves conjugacy classes. Two years later he answered his
own question by constructing a family of such groups. Using the small group
library in MAGMA we determine all of the groups of order n < 512 that possess
such an automorphism. Our investigations led to the discovery of four new
infinite families of such groups, all of which are 2-groups of coclass 4.
