The purpose of this paper is to state some new results on an -nilpotent group, which was introduced by Barzegar and the second author, for any fixed auto- morphism of a group G. They investigated some properties of an -nilpotent group and proved that an -nilpotent group is nilpotent but the converse is not valid in gen- eral. Therefore, authors tried to prepare some conditions that under what conditions a nilpotent group is -nilpotent for an automorphism . A group G is said to be non-inner nilpotent, whenever it is nilpotent related to a non-inner automorphism of the group G. In this paper, we provide some examples of non-inner nilpotent groups, also we prove that every nilpotent group of maximal class is non-inner nilpotent and we try to classify all finite non-abelian non-inner nilpotent groups of order pn, n 5, for an odd prime p. Central automorphisms fixing the center of a group element-wise may play an important role to this classification.

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