A quantization of the symmetry $G$ (if it exists) is a projective representation $\rho:G\rightarrow U(\mathbb P\mathcal H)$ into the unitary operators on $\mathbb P\mathcal H$.
To clarify - $\mathbb P\mathcal H$ is not a vector space, and so I'm not so sure what a unitary operator on $\mathbb P\mathcal H$ would be. My assumption is that you are referring to the ray product $(\Phi,\Psi)$ between elements of $\mathbb P\mathcal H$, defined via
$$(\Phi,\Psi):= \frac{|\langle\phi,\psi\rangle|}{\Vert\phi\Vert \ \Vert\psi\Vert}$$
where $\phi,\psi\in \mathcal H$ are any two representatives of the equivalence classes $\Phi$ and $\Psi$. A symmetry transformation is then a map $T:\mathbb P\mathcal H \rightarrow \mathbb P\mathcal H$ such that $(T\Phi,T\Psi)=(\Phi,\Psi)$. This looks rather like the definition of a unitary operator on a Hilbert space, but there is no notion of linearity present here.
Wigner's theorem tells us that any such $T$ can be induced by some unitary or antiunitary operator on $\mathcal H$, which descends to a transformation on $\mathbb P\mathcal H$ in the obvious way. By extension, if we want to represent the action of a symmetry group $G$ rather than just a single transformation, we might seek linear representations $\rho:G\rightarrow U(\mathcal H)$. However, requiring $\rho$ to be a true group homomorphism is generally too strong; if it is a homomorphism up to a phase - i.e. $\rho(g)\rho(h)=\rho(gh)C(g,h)$ for some phase factor $C(g,h)$ - then it also induces a well-defined group action on $\mathbb P\mathcal H$ because the extra phase factor is lost during the projectivization from $\mathcal H$ to $\mathbb P\mathcal H$.
The takeaway is that the action of a symmetry group $G$ on $\mathbb P\mathcal H$ can be understood through a projective unitary representation of $G$ on $\mathcal H$, so we need to understand the latter if we want to understand how to implement symmetry groups in our theories.
From what I have read, it seems like we want (or physically expect?) this $\rho$ to lift to a representation $\hat{\rho}: G \rightarrow U \left( \mathcal{H} \right)$
If $\rho:G\rightarrow \mathrm{Aut}(\mathbb P\mathcal H)$ is a group action whose elements are symmetry transformations as defined above, then Wigner's theorem tells us that that it lifts to a projective unitary representation $\hat \rho:G\rightarrow \mathbb PU(\mathcal H)$. It may lift to a genuine representation - i.e. one might be able to arrange that the phase factors $C(g,h)$ referenced above are all equal to $1$ - but in general this cannot be done.
I am trying to understand why, in certain circumstances, we quantize the central extension of $G$, rather than $G$ itself.
As it turns out, projective representations of $G$ can be put into one-to-one correspondence with (genuine) linear representations of its central extensions (the same is true for the Lie algebra $\mathfrak g$). Since linear representations are far nicer to work with, this is what we tend to study.
ACuriousMind has written a wonderful answer on this subject which you can find here. I doubt any summary I could provide would do it justice, so I suggest reading it directly if you haven't already done so.
