According to [Djokovic and Maizan][1], the Specht module $V_{(3, 1, 1)}$ of $S_5$ is monomial. This is a representation of dimension $6$, induced from a representation of dimension $3$ of $A_5$. Since $A_5$ has no subgroup of index $3$ (see [here][2] for example), this representation of $A_5$ cannot be monomial.


  [1]: http://dx.doi.org/10.1016/0021-8693(75)90041-1
  [2]: http://groupprops.subwiki.org/wiki/Subgroup_structure_of_alternating_group:A5