All the permutations that commute with (1 2)(3 4 5) in $S_5$

Question

Determine all the permutations $\sigma \in S_5$ that commute with the permutation $\alpha = (1 \space 2)(3 \space 4 \space 5).$

Answer 1

Hint: $\varphi\in S_n$ then $$\varphi(a_1,a_2,\cdots,a_k)\varphi^{-1}=(\varphi(a_1),\cdots,\varphi(a_k)).$$

Suppose $\varphi\in S_5$ such that $\varphi\sigma \varphi^{-1}=\sigma$ then $\varphi(1)=1,2$ and hence $\varphi(2)=2,1$.

Similarly for $(3,4,5)$.