Sylow intersections and control of fusion

Let G be a finite group and let P be a Sylow p-subgroup of G. We prove that if the automizer of each intersection of two distinct Sylow p-subgroups is a p-group, then NG(P)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{N}\,}}_G(P)$$\end{document} controls strong G-fusion in P. In fact, more general results will be discussed in this note.