Semisimplicity of the induced representation of an irreducible representation Let $G$ be an arbitrary group, $H$ be a subgroup of finite index $n$ and $k$ be an algebraically closed field of characteristic prime to $n$.
Suppose that we have an irreducible representation
$$\rho: H\to \mathrm{GL}(V)$$
where $V$ is a finite-dimensional $k$-vector space.

Is the induced representation $\mathrm{Ind}_H^G(\rho)$ semisimple?

This will certainly be true if $H$ is finite of order prime to the characteristic, or if we are in characteristic $0$ and $G$ is compact. Can we get away with less in this case?
If $\mathrm{Ind}_H^G(\rho)$ is not semisimple, does the situation change if we assume that $\rho$ is actually the restriction of some irreducible representation $\sigma: G\to\mathrm{GL}(V)$?
Edit: Alternatively, would the situation change if we knew that $H$ was normal in $G$?
 A: Let $k$ have characteristic $p$ and $|G:H|$ prime to $p$. Even inducing the trivial representation of $H$ (which is certainly the restriction of an irreducible representation!) to $G$ does not in general give a semisimple representation.
For example, take $p=2$, $G=A_5$ the alternating group of degree $5$, and $H$ a Sylow $2$-subgroup. Then inducing the trivial representation gives a non-semisimple module.
If $H$ is normal in $G$, then the induced representation is always semisimple. 
Let $V$ be an irreducible $kH$-module, and let $\uparrow$ and $\downarrow$ denote induction and restriction between H and G. Then $V\!\!\uparrow\downarrow$, is semisimple, as it's the direct sum of irreducible $kH$-modules $V\otimes g$, where $g$ runs over a set of coset representatives of $H$ in $G$.
Suppose $U$ is a $kG$-submodule of $V\!\!\uparrow$. As a $kH$-module it is a direct summand, so there is a $kH$-module homomorphism $\alpha:V\!\!\uparrow\to U$ projecting onto $U$. As in the usual proof of Maschke's Theorem, 
$$\tilde\alpha(v)=\frac{1}{|G:H|}\sum_g\alpha(vg)g^{-1},$$
where the sum is over a set of coset representatives, is a $kG$-module homomorphism $\tilde{\alpha}:V\!\!\uparrow\to U$ projecting onto $U$. So $U$ is a $kG$-module direct summand of $V\!\!\uparrow$.
Since every $kG$-submodule of $V\!\!\uparrow$ is a direct summand, $V\!\!\uparrow$ is a semisimple $kG$-module.
