(Here by $\mathrm{SL}(2,3)$, I am referring to $2 \times 2$ matrices of determinant $1$ over the field of order $3$, and by $\mathrm{GL}(n, \mathbb{C})$, I am referring to $n \times n$ nonsingular matrices over the complex numbers. The $\mathrm{P}$ prefix refers to quotienting one of these groups by its center, the subgroup of scalar matrices.)
In learning about projective representations and Schur multipliers in group theory, I am attempting to prove that $\mathrm{SL}(2,3)$ is a cover of $\mathrm{PSL}(2,3)$. The last step is to show that the “projective lifting property” holds:
Let $v \colon \mathrm{SL}(2,3) \to \mathrm{PSL}(2,3)$ and $\pi \colon \mathrm{GL}(n,\mathbb{C}) \to \mathrm{PGL}(n,\mathbb{C})$ be the quotient maps. Then, for any homomorphism $\tau \colon \mathrm{PSL}(2,3) \to \mathrm{PGL}(n,\mathbb{C})$, there exists a homomorphism $\overline{\tau} \colon \mathrm{SL}(2,3) \to \mathrm{GL}(n,\mathbb{C})$ such that $\pi \overline{\tau} = \tau v$. $$ \require{AMScd} \begin{CD} \mathrm{SL}(2, 3) @>{\overline{\tau}}>> \mathrm{GL}(n, \mathbb{C}) \\ @V{v}VV @VV{\pi}V \\ \mathrm{PSL}(2, 3) @>>{\tau}> \mathrm{PGL}(n, \mathbb{C}) \end{CD} $$
From the existence of a cover, we would then be able to immediately conclude what the Schur multiplier of $\mathrm{PSL}(2,3)$ is: it is isomorphic to the subgroup of scalar matrices of $\mathrm{SL}(2,3)$, which is in turn isomorphic to $\mathbb{Z}_2$.
I am looking for a proof (using methods that a reader of Rotman’s An Introduction to the Theory of Groups can follow) that the highlighted “projective lifting property” above holds.