Quotient-coprime automorphism-faithful subgroup

Definition
A subgroup $$H$$ of a finite group $$G$$ is termed quotient-coprime automorphism-faithful if $$H$$ is a normal subgroup of $$G$$, and:

If $$\varphi$$ is a non-identity automorphism of $$G$$ whose order is relatively prime to the order of $$G$$, such that $$\varphi(H) = H$$, then the induced map by $$\varphi$$ on $$G/H$$ is also a non-identity automorphism.

Facts

 * Frattini subgroup of finite group is quotient-coprime automorphism-faithful

Weaker properties

 * Stronger than::Normal subgroup