Subgroup-cofactorial automorphism-invariant subgroup

For a finite subgroup
Suppose $$G$$ is a group and $$H$$ is a finite subgroup. $$H$$ is termed a subgroup-cofactorial automorphism-invariant subgroup of $$G$$ if $$H$$ is invariant under every automorphism $$\sigma$$ of $$G$$ of finite order for which every prime divisor of the order of $$\sigma$$ is a prime divisor of the order of $$H$$.

For a periodic subgroup
Suppose $$G$$ is a group and $$H$$ is a periodic subgroup, i.e., a subgroup in which every element has finite order. $$H$$ is termed a subgroup-cofactorial automorphism-invariant subgroup of $$G$$ if $$H$$ is invariant under every automorphism $$\sigma$$ of $$G$$ of finite order for which every prime divisor of the order of $$\sigma$$ is a prime divisor of the order of some element of $$H$$.

For a subgroup that is not periodic
If $$H$$ is a subgroup of $$G$$ with an element of infinite order,, we declare $$H$$ to be subgroup-cofactorial automorphism-invariant if and only if $$H$$ is a characteristic subgroup of $$G$$.

Stronger properties

 * Weaker than::Characteristic subgroup
 * Weaker than::Cofactorial automorphism-invariant subgroup
 * Weaker than::Sub-cofactorial automorphism-invariant subgroup

Weaker properties

 * Stronger than::Left-transitively 2-subnormal subgroup:
 * Stronger than::2-subnormal subgroup
 * Stronger than::Subnormal subgroup

Incomparable properties

 * Normal subgroup: