Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e. \begin{equation} D(g) D(h) = e^{i \omega(g,h)} D(gh) \end{equation} These can be classified by the equivalence relation $\omega(g,h) \sim \omega(g,h)+\theta(g)+ \theta(h) - \theta(gh)$ subject to the condition $\omega(g,h)+\omega(gh,l)-\omega(h,l)-\omega(g,hl) = 0$. The distinct equivalent classes of projective representations are labeled by elements of $H^2(G,U(1))$. I know that for every such finite group, there is atleast one finite covering group C with the property that every projective representation of G can be lifted to an ordinary representation of C [1].

My question is about the relationship between the irreducible representations (irreps) of C and the group $H^2(G,U(1))$. Specifically, is the following statement true?:

Every irrep $\Gamma_i$ of C can be associated an element of $\nu \in H^2(G,U(1))$ like $\Gamma^\nu_i$. The group property of $H^2(G,U(1))$ is reflected in the Clebsch-Gordan decomposition of tensor product of irreps of C: \begin{equation} \Gamma^\nu_i \otimes \Gamma^\mu_j \cong \bigoplus_k \Gamma^{\nu+\mu}_k \end{equation}

I have noticed that this is true for all cases I have seen when $ H^2(G,U(1)) \cong \mathbb{Z}_2$ like $G = \mathbb{Z}_2 \times \mathbb{Z}_2$, $C = D_8$ but I am unsure if this is true in general.

[1] https://en.wikipedia.org/wiki/Schur_multiplier#Relation_to_projective_representations

PS: I am a physicist and my interest in the above question comes from its use in condensed matter physics.