Characteristic subgroup of Sylow subgroup

Definition
A subgroup of a finite group is termed a characteristic subgroup of Sylow subgroup if it can be expressed as a characteristic subgroup of some Sylow subgroup.

Stronger properties

 * Weaker than::Sylow subgroup

Weaker properties

 * Stronger than::Automorph-conjugate subgroup

Facts

 * Frattini's argument, which is generally stated for Sylow subgroups, holds for all automorph-conjugate subgroups. In particular, it holds for characteristic subgroups of Sylow subgroups. In words: if $$K \le L \le G$$ are such that $$K$$ is a characteristic subgroup of a Sylow subgroup of $$L$$, and $$L$$ is normal in $$G$$, then $$G = LN_G(K)$$.