All permutations from $S_6$ and $S_7$ by which $(1,2)(3,4,5)$ is conjugate to itself

Question

The task is to find all permutation $\tau$ from $S_6$ and $S_7$ such that: $$\tau^{-1}(12)(345)\tau=(12)(345)$$

I think the answer is: $\{id \in S_6 , id \in S_7 , (67) \in S_7\}$

I would just like to confirm as I am not sure if I understand this topic correctly.

Answer 1

Recall that $$\tau ^{-1}(12)(345)\tau=\tau^{-1}(12)\tau \tau^{-1}(345)\tau=(\tau (1)\,\tau( 2))\,(\tau (3)\,\tau (4)\,\tau (5))$$

Thus you need $$(\tau (1)\,\tau (2))\,(\tau (3)\,\tau (4)\,\tau (5))=(12)(345)$$ Does this give you an idea?