Lifting representations, kernels and invariant subspaces 
Let $G$ be a group, $N \triangleleft G$, $G/N$ the corresponding quotient group.
Suppose $\rho : G/N \longrightarrow GL(\mathbb{C})$ is a representation of $G/N$. Then the composition
$$\overset{\sim} \rho : G \longrightarrow G/N \longrightarrow GL(\mathbb{C})$$
$$g \mapsto gN \mapsto \cdot$$
is a representation of $G$ lifted to $G$. Note this is a representation on the same vector space $\mathbb{C}^n$.
It has the same action and the same invariant subspaces which implies that $\rho$ irreducible $\iff \overset{\sim} \rho$ irreducible.

I cannot see why or how to prove that it will have the same invariant subspaces, I can roughly see how it would have the same action.

Note also that $\ker \rho > N$. Conversely, if $\sigma$ is a representation of $G$ and $\ker \rho > N$, then $\sigma$ is lifted from a unique representation of $G/N$. [namely $gN \mapsto \sigma(g)$]

I think this follows from one of the isomorphisms theorem but again I cannot see how it would work exactly.
 A: For the first part. Let $\rho:G/N\rightarrow GL_n(\mathbb{C})$ be given. Then $W\subseteq GL_n(\mathbb{C})$ is a $G/N,\rho$-invariant subspace if and only if $\rho(gN)(W)\subseteq W$ for all $gN\in G/N$ if and only if $\tilde{\rho}(g)(W)\subseteq W$ (by definition of $\tilde{\rho}$) for all $g\in G$ if and only if $W$ is a $G,\tilde{\rho}$-invariant subspace. From this equivalence $\rho$ and $\tilde{\rho}$ have the same invariant subspaces. In particular, irreducibility of one is equivalent to irreducibility of the other one.
For the second part, the property you are looking for is this one. For any group $H_1, H_2$, $f$ a group morphism from $H_1$ to $H_2$ and $N$ a normal subgroup of $H_1$. Let us denote $\pi:H_1\rightarrow H_1/N$ the projection. Then there exists a group morphism $g:H_1/N\rightarrow H_2$ such that :
$$f=g\circ \pi \text{ i.e. } \pi\text{ factorizes } f$$
If and only if $Ker(f)$ contains $N$. 
To apply this to your setting take $H_1:=G$, $N:=N$, $H_2:=GL_n(\mathbb{C})$ and $f:=\rho$ your representation.
