Criterion for projective representation to lift to linear representation

Statement
Suppose $$G$$ is a finite group and $$N$$ is a central subgroup of $$G$$. Suppose $$Q = G/N$$ and $$\rho: Q \to PGL_d(\mathbb{C})$$ is a homomorphism of groups, and hence a projective representation of $$G/N$$. Let $$\alpha:G \to Q$$ and $$\pi_d:GL_d(\mathbb{C}) \to PGL_d(\mathbb{C})$$ be the obvious quotient maps. We call a linear representation $$\theta: G \to GL_d(\mathbb{C})$$ a lift of $$\rho$$ if $$\pi_d \circ \theta = \rho \circ \alpha$$.

The statement is that there exists a linear representation that is a lift of $$\rho$$ if and only if the following holds.

Description of condition in cohomology language
Consider the following biadditive map:

$$H^2(Q;N) \times \operatorname{Hom}(N,\mathbb{C}^*) \to H^2(Q;\mathbb{C}^*)$$

Fixing the element of $$H^2(Q;N)$$ corresponding to the extension group $$G$$, we get a homomorphism:

$$\operatorname{Hom}(N,\mathbb{C}^*) \to H^2(Q;\mathbb{C}^*)$$

from the group of one-dimensional representations of the central subgroup $$N$$ to $$H^2(Q;\mathbb{C}^*)$$.

The projective representation $$\rho$$ lifts to a linear representation if and only if the image of the above homomorphism contains the cohomology class (i.e., the element of $$H^2(Q;\mathbb{C}^*)$$ corresponding to $$\rho$$.

Description of condition in homology language
Consider the formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization:

$$0 \to \operatorname{Ext}^1(Q^{\operatorname{ab}},N) \to H^2(Q;N) \to \operatorname{Hom}(H_2(Q;\mathbb{Z}),N) \to 0$$

The extension group $$G$$ corresponds to an element of $$H^2(Q;N)$$, which hence maps to an element of $$\operatorname{Hom}(H_2(Q;\mathbb{Z}),N)$$. By composition, this defines a map:

$$\operatorname{Hom}(N,\mathbb{C}^*) \to \operatorname{Hom}(H_2(Q;\mathbb{Z}),\mathbb{C}^*) = H^2(Q;\mathbb{C}^*)$$

The condition is that the image of this homomorphism should contain the cohomology class corresponding to the projective representation.