When does a representation of $H\subset G$ on $V$ extend to a representation of $G$ on $V$?

Question

Let $G$ be a finite group, $H$ a subgroup, and $\varphi:H\rightarrow GL(V)$ a finite-dimensional representation of $H$ over a characteristic zero, algebraically closed field. Let $\chi$ be the character of this representation.

If $\chi$ takes different values on two different conjugacy classes in $H$ that become conjugate in $G$, then clearly $\varphi$ cannot extend to a representation of $G$ on $V$.

Is this the only obstruction?

I.e. if $\chi$ is the restriction of a class function on $G$, is $\varphi$ the restriction of a representation of $G$ on $V$?

Another way to ask the question is, with this assumption about $\chi$, does there exist a class function that extends $\chi$ and is equal to an $\mathbb{N}$-linear combination of irreducible characters of $G$?

Or are there other things that can go wrong? And if there are, are they straightforward to test for, or is there some whole subtle cohomological theory of obstructions, or what?

Thanks in advance!

Answer 1

Here is an example where it cannot extend. Let $H$ be a subgroup of order $2$ that lies in the commutator subgroup of $G$ (for example a subgroup of order $2$ in $A_4$) and let $\phi$ be the nontrivial linear representation of $H$ that maps the element of order $2$ to $-I$. Then, since $H \le [G,G]$, $H$ lies in the kernel of any linear representation of $G$, so $\phi$ cannot extend to a representation of $G$.

In geenral you cannot expect representations to extend in this way.

Answer 2

The example you gave about $H$ having two conjugacy classes which become conjugate in $G$ is wrong. We can actually still find a value of $\chi\uparrow G$. (This is read "$\chi$ induced to $G$").

Let's say $x_1^H$ and $x_2^H$ are two conjugacy classes in $H$ which become a single conjugacy class $x^G$ in $G$. Then the formula for $g\in x^G$ is

$$\chi(g)=|C_G(g)|\left(\frac{\chi(x_1)}{|C_H(x_1)|}+\frac{\chi(x_2)}{|C_H(x_2)|}\right)$$