Prove that for each $\displaystyle \sigma \in Aut(S_n)$

$\displaystyle \sigma: (1 \, 2) \mapsto (a \, b_2), \, \sigma: (1 \, 3) \mapsto (a \, b_3), \ldots, \sigma: (1,\, n) \mapsto (a \, b_n)$.

I know that an automorphism sends a transposition to a transposition so $\displaystyle \sigma: (1 \, 2) \mapsto (a \, b_2)$ is fine. How to prove the rest. I tried to use contradiction. denote $\displaystyle \sigma((1 \, 3))= \tau$. Assume $\displaystyle \tau (a)=a, \tau (b_2)=b_2$. Then we should get a contradiction but i couldn't arrive at one.