If $ G $ is a finite soluble group and satisfies the permutizer condition, then for any odd prime $ p $, $ G $ is $ p $-supersoluble.

Question

If $ G $ is a finite soluble group and satisfies the permutizer condition, then for any odd prime $ p $, $ G $ is $ p $-supersoluble. It is for me problem that why $ G $ can't $ 2 $-supersoluble group. ( If for any proper subgroup $ H $ of $ G $, $ H < P_{G}(H) $ such that $ P_{G}(H)=\langle g\in G \ \vert \langle g \rangle H = H \langle g \rangle \rangle $, then said $ G $ satisfying the permutizer condition.)