Group Actions · Medium problem

Let $H\le S_4$ be the cyclic subgroup of order $4$ generated by the $4$-cycle $(1\,2\,3\,4)$. $S_4$ acts on the set of left cosets of $H$ by left multiplication: $g\cdot(xH)=(gx)H$. Find $|\operatorname{Stab}(eH)|$, the stabilizer of the coset $eH=H$ itself.