Let $N$ be a normal subgroup of a finite group $G$. Then for each $g \in G$ there exists an automorphism of $N$ given by $\phi_g(n)=g^{-1}ng$. Suppose we impose the following condition: $$\phi_g(n) \neq n^{-1},~~~\forall g\in G,~~~ \forall n \in N \backslash \{1\}.$$ What I can show so far is that for this condition to hold, it is necessary that $N$ have odd order. If $N$ is contained in the center of $G$, this is also sufficient.
My question is that whether you have seen such a condition before in group theory literature, and if not, what would be an appropriate terminology to use for such a normal subgroup?