Comparable with all normal subgroups implies characteristic in finite nilpotent group

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite nilpotent group. That is, it states that in a Finite nilpotent group (?), every subgroup satisfying the first subgroup property (i.e., Subgroup comparable with all normal subgroups (?)) must also satisfy the second subgroup property (i.e., Characteristic subgroup (?)). In other words, every subgroup comparable with all normal subgroups of finite nilpotent group is a characteristic subgroup of finite nilpotent group.
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Statement

If a subgroup of a finite nilpotent group is comparable with all normal subgroups, then it is a characteristic subgroup.

Facts used

1. Comparable with all normal subgroups implies normal in nilpotent
2. Normal subgroup comparable with all normal subgroups implies characteristic in finite

Proof

The proof follows by combining facts (1) and (2).