Group Actions · Hard problem

$S_4$ acts on the $3$ ways to partition $\{1,2,3,4\}$ into two $2$-element subsets: $\{\{1,2\},\{3,4\}\}$, $\{\{1,3\},\{2,4\}\}$, $\{\{1,4\},\{2,3\}\}$. What is the order of the kernel of this action?