Characterization of $\tau \in S_n$ such that $\sigma:=(1 \space 2)\tau$ commutes with two distinct transpositions

Question

Possibly ancillary to another problem, I want to characterize the permutations $\tau \in S_n$ such that $\sigma:=(1 \space 2)\tau$ commutes with two distinct transpositions. If relevant, we can use the constraint that $\tau$ mustn't be an $n$-cycle.