Construct representation of $G$ from representation of $G/N$

Question

Assume $G$ is a random finite group, and $N$ is a normal subgroup of $G$.
How do I construct a representation of $G$ from an existing representation $G/N$?

So far I have written: $\varphi:G/N\rightarrow GL(V)\\ \psi:G\rightarrow GL(W)$ with $V,W$ vector spaces. Then we need to construct $\psi$ from $\varphi$.
I also wrote this, however I do not know if I can do anything with this. $$\begin{align} &G/N\xrightarrow{\varphi}GL(V)\\ &\downarrow\hspace{2cm}\downarrow\\ &\hspace{0.1cm}G\hspace{0.5cm}\xrightarrow{\psi}GL(W) \end{align}$$

Answer 1

That's a very natural definition. Define $\psi:G\to GL(V)$ by $\psi(g)=\varphi(gN)$. It is easy to check that this is a homomorphism of groups, and thus a representation of $G$.

Answer 2

The composition $G \to G/N \xrightarrow{\varphi} GL(V)$ is the desired representation, where $G \to G/N$ is the canonical projection.