We consider $\mathbb C$-linear representations of finite groups.
If $H$ is a subgroup of $G$, and $(W,\theta)$ is an $H$-representation, irreducibility Mackey's criterion states that $Ind_H^G W$ (the $G$-representation induced by $W$ on $G$) is irreducible if and only if $W$ is irreducible and the following condition holds: for each $x\in G, x\not \in H$, the $H$-representation $W^x$ defined on $xHx^{-1}\cap H$ as $(W,\theta^x)$, with $\theta^x(h)=\theta(x^{-1}hx)$ has no subrepresentations in common with the restriction of $(W,\theta)$ to $H\cap H^x$.
I understand the proof of the criterion, via the Frobenius-recipocity, but I would like to get more intuition about it. For example, is it possible to give an explicit subrepresentation of $V$ in case there is such a a common subrepresentation of $W$ and $W^x$? Or any other hint to help me with intuition? Thank you all.