According to Djokovic and Maizan, 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 for example), this representation cannot be monomial.
Amritanshu Prasad
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