Let $G_1$ and $G_2$ be finite groups such that $G_1$ is a subgroup of $G_2$. Let $V$ be a representation of $G_1$ (over some field; I am not assuming that the characteristic of the field is $0$ or that $V$ is finite dimensional). There is a natural $G_1$-equivariant map $\phi : V \rightarrow \text{Ind}_{G_1}^{G_2} V$. Let $W$ be a $G_2$-subrepresentation of $\text{Ind}_{G_1}^{G_2} V$ such that $W \neq \{0\}$.
Question : How do you show that $\phi(V) \cap W \neq \{0\}$?