For Every Representation of a Subgroup is there a Group such that the Group Representation is Irreducible? Let $H$ be a finite group, Let $V$ be a finite dimensional vector space over $\mathbb{C}$, and $\rho$ a representation of $H$ on $V$. Is there, for any representation $\rho$ of $H$ on $V$, another finite group $G$, $H < G$, such that $\tau$ is a represenation of $G$ on $V$, $\tau(h) = \rho(h) \ \forall h \in H$, and $\tau$ is irreducible?
My intuition says this is false, but I have had no traction whatsoever in proving it.
 A: The question for general groups has been answered in the positive.
If $H$ is a finite group and you insist that $G$ is also a finite group, however, then the answer is no, in general.
Let $H = 2.A_5 \times \mathbf{Z}/p \mathbf{Z}$ for any prime $p > 7$. (Here $2.A_5 = \mathrm{SL}_2(\mathbf{F}_5)$ is a non-trivial central extension of $A_5$.) The group $2.A_5$ has a faithful representation of dimension $2$, and $\mathbf{Z}/p \mathbf{Z}$ has a faithful representation of dimension $1$. Let $V$ be the direct sum of these representations, which is reducible of dimension $3$.
Then $V$ as a representation of $H$ is not the restriction of any irreducible
representation of a finite group $G$. We may assume that $G$ acts faithfully on $V$, since otherwise we
can replace $G$ by the image, which still contains $H$ because $H$ acts faithfully.
I claim that the action of $G$ on $V$ is  primitive,  that is, it is not induced from a  subgroup $P$ with $[G:P] > 1$.
Since $3 = \mathrm{dim}(V)$ is prime, $V$ would necessarily be induced from a character $\chi$ of an index $3$ subgroup $P$. But this would force $G$
to be solvable: it admits a map to $S_3$ (acting on the left cosets $G/P$), and the restriction of $V$ to the kernel $N$ is just a direct sum of characters, namely $\chi |_{N}$ and its conjugates by $G$. Since $N$ acts faithfully, this means that $N$ is abelian, and thus $G$ is solvable. Since $H$ is not solvable, this is impossible. On the other hand, there are only finitely many subgroups of $\mathrm{GL}_n(\mathbf{C})$ which act primitively.
For $n = 3$, they have projective image one of six groups, the non-simple examples being $A_5$, $A_6$, and $\mathrm{PSL}_2(\mathbf{F}_7)$.
(See finite subgroups of PGL(3,C))
Since the element of order $p$ in $H$ does not act diagonally on $V$, it acts non-trivially on the projective representation. For $p$ big enough ($p > 7$ in this case) this is a contradiction
with the possible primitive subgroups of $\mathrm{PGL}_3(\mathbf{C})$.
