Subgroup whose normalizer in the Sylow is Sylow in the normalizer

Definition
Suppose $$G$$ is a finite group and $$p$$ is a prime number. Suppose $$S$$ is a $$p$$-Sylow subgroup, and $$U$$ is a subgroup of $$S$$. We say that $$U$$ satisfies the property that its normalizer in the Sylow is Sylow in the normalizer if $$N_S(U)$$ is a $$p$$-Sylow subgroup of $$N_G(U)$$.

Stronger properties

 * Weaker than::Well-placed subgroup
 * Weaker than::Tame Sylow intersection: Here, the tame intersection is viewed as a subgroup of the first Sylow subgroup being intersected.

Facts

 * Every p-subgroup is conjugate to a p-subgroup whose normalizer in the Sylow is Sylow in its normalizer: This means that the property tells us nothing about $$U$$ or $$S$$ in isolation but only provides information about the way $$U$$ is embedded in S relative to $$G$$.