Group Actions · Hard problem

$S_5$ acts on the $10$ two-element subsets of $\{1,2,3,4,5\}$ by $\sigma\cdot\{a,b\}=\{\sigma(a),\sigma(b)\}$. How many of the $10$ subsets are fixed (setwise) by the permutation $\sigma=(1\,2\,3)$?