Let $H\trianglelefteq G$ (finite groups) and let $V$ be a (complex) representation of $G$, and let $W$ be a $H$ subrepresentation of the restriction of $V$ to $H$.
Does exist an irreducible $G-$subrepresentation of $V$ such that contains $W$? (i know that exists an irreducible $G-$subrepresentation of $V$ such that contains a $H-$subrepresentation isomorphic to $W$ but I want it to contain $W$)
No, take H to be the trivial group and let W be the span of any vector in V. – Jonathan 21 hours ago