Let \mathcal{S} be the set of permutations of \{1,2, \ldots, 6\}, and let \mathcal{T} be the set of permutations of \mathcal{S} that preserve compositions: i.e., if F \in \mathcal{T}, then
F\left(f_{2} \circ f_{1}\right)=F\left(f_{2}\right) \circ F\left(f_{1}\right)\tag{A.22}
for all f_{1}, f_{2} \in \mathcal{S}. Find the number of elements F \in \mathcal{T} such that if f \in S satisfies f(1)=2 and f(2)=1, then (F(f))(1)=2 and (F(f))(2)=1.