Finite groups with few conjugacy classes of $p$-subgroups

Let G be a finite group, and let p be an odd prime of |G| and Sylow p -subgroups of G be non-abelian. In this paper, we prove that any two p -subgroups of equal order in G are conjugate if and only if G/O_{p'}(G) is isomorphic to {}^{2}F_{4}(2) and p=3 , \mathrm{Ru} and p=3 , J_{4} and p=3 , \mathrm{Th} and p=5 , or an almost simple group of the socle {}^{2}F_{4}(2^{2n+1}) with n > 0, n\not\equiv 1\bmod 3 and p=3 . This solved a problem of Brandl’s.