Mackey's test for irreducibility of induced representation over $\mathbb{C}$ is:


Let $G$ be a finite group, $H\leq G$, $W$ be a representation of $H$, and $W^x$ be conjugate representation of $H^x=xHx^{-1}$. Then following are equivalent:


(i) $Ind^G\_H(W)$ is irreducible.


(ii) $W$ is irreducible and for each $x\in G\setminus H$, $W$ and $W^x$ have no common irreducible component, when restricted to $H\cap H^x$. 


There is [question](https://mathoverflow.net/questions/84510/mackey-irreducibility-criteria) posted, about changing the ground field, and the answer posted is (are) "Yes". 


But, $(ii)\Rightarrow (i)$ is true for any field of characteristic zero or prime to $|G|$; how does $(i)\Rightarrow (ii)$ for such fields? 



Irreducibility of Induced Representation
No. Over $\mathbb R$ let $G$ be the quaternion group of order $8$, $H$ the subgroup of order $2$, $W$ the nontrivial one-dimensional representation.


EDIT For an even simpler example, see Kevin Ventullo's comment!


