Let $H$ be a subgroup of $G$. Let $\rho$ be an irr representation of $G$ induced from an irr representation $\theta$ of $H$. It is well known that $\rho$ is monomial if $\theta$ is monomial. Is it possible that $\rho$ is monomial but $\theta$ is not monomial? 


In other words, can monomial representations induced from nonmonomial representations?



Can monomial representations induced from nonmonomial representations?
According to [Djokovic and Maizan](http://dx.doi.org/10.1016/0021-8693(75)90041-1), 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](http://groupprops.subwiki.org/wiki/Subgroup_structure_of_alternating_group:A5) for example), this representation of $A\_5$ cannot be monomial.


