 This is essentially because the infinitary permutation takes permutations of finite supports to permutations of finite support.For finite groups, the classical example is a group of order 32 obtained as the semidirect product of the cyclic ring on 8 elements, by its group of units acting via multiplication.Finding a class automorphism in the stability group that is not inner boils down to finding a cocycle for the action that is locally a coboundary but is not a global coboundary.