Exhibit two distinct Sylow 2-subgroups of $S_5$ and an element of $S_5$ that conjugates one into the other.
Sketch of my answer:
$p=2,\, \alpha=3,\, m=15, n_2=1,3,5,15 $ and so the number of Sylow 2-subgroups of $S_5$ are $15.$ Two distinct Sylow 2-subgroups of $S_5$ are $<(12)>$ and $<(23)>.$ Observe that $(123)\in S_5$ and $(123)(12)(132)=(23).$ So $(123)$ is the element that conjugates one into the other.
Please comment/correct my answer.