Group Actions · Hard problem

$S_4$ acts on the $3$ ways to split $\{1,2,3,4\}$ into two unordered pairs – $\{\{1,2\},\{3,4\}\}$, $\{\{1,3\},\{2,4\}\}$, $\{\{1,4\},\{2,3\}\}$ – by applying $\sigma$ to every element. How many permutations in $S_4$ send the split $\{\{1,2\},\{3,4\}\}$ to itself (as a set of two blocks, possibly swapping the two blocks)?