Group having an automorphism whose restriction to the center is the inverse map

Definition
A group $$G$$ is a group having an automorphism whose restriction to the center is the inverse map if there exists an automorphism $$\sigma$$ of $$G$$ such that the restriction of $$\sigma$$ to the center $$Z(G)$$ is the automorphism $$g \mapsto g^{-1}$$.

Stronger properties

 * Weaker than::Abelian group
 * Weaker than::Centerless group
 * Weaker than::Group whose center is a direct factor
 * Weaker than::Group whose center has order two
 * Weaker than::Group whose center is an AEP-subgroup
 * Weaker than::Group having a class-inverting automorphism