Sylow implies MWNSCDIN

Statement
Any Sylow subgroup of a group is a MWNSCDIN-subgroup.

Sylow subgroup
A subgroup $$P$$ of a group $$G$$ is termed a Sylow subgroup if its order is a power of a prime $$p$$ and the index of $$P$$ in $$G$$ is relatively prime to $$p$$.

MWNSCDIN-subgroup
A subgroup $$H$$ of a group $$G$$ is termed a MWNSCDIN-subgroup if, given a collection of normal subsets $$A_i, i \in I$$ and $$B_i, i \in I$$ of $$H$$, such that there exists $$g \in G$$ such that $$gA_ig^{-1} = B_i$$, for all $$i \in I$$, then there exists a $$h \in N_G(H)$$ such that $$hA_ih^{-1} = B_i$$ for each $$i \in I$$.

Similar facts

 * Sylow implies WNSCDIN
 * Pronormal implies MWNSCDIN
 * Pronormal implies WNSCDIN
 * Sylow and TI implies CDIN
 * Abelian Sylow implies SCDIN

Opposite facts

 * Sylow not implies CDIN

Facts used

 * 1) uses::Sylow implies pronormal
 * 2) uses::Pronormal implies MWNSCDIN

Proof using given facts
The proof follows from facts (1) and (2).