Irreducible representation of quotient group composed with quotient map gives irreducible representation of group

Statement
Suppose $$G$$ is a group, $$H$$ is a normal subgroup of $$G$$, $$L = G/H$$, and $$\pi:G \to L$$ is the quotient map. Suppose $$\alpha$$ is an fact about::irreducible linear representation of $$L$$ over a field $$F$$, i.e., $$\alpha$$ is a homomorphism $$L \to GL(V)$$ where $$V$$ is a $$F$$-vector space.

Then, the map $$\alpha \circ \pi:G \to GL(V)$$ is an irreducible representation of $$G$$ over $$F$$.

Applications

 * Degrees of irreducible representations of quotient group are contained in degrees of irreducible representations of group