Exercise 6.6.5 Weibel Homological Algebra For any field $k$ and any $n\in \mathbb{N}$, let $\gamma$ denote the class in $H^{2}(PGL_{n}(k);k^{*})$ corresponding to the extension
$$1 \rightarrow k^{*} \rightarrow GL_{n}(k) \rightarrow PGL_{n}(k) \rightarrow 1$$
Let $\rho : G \rightarrow PGL_{n}(k)$ be a projective representation. Then we need to show that $\rho$ lifts to a linear representation $\rho': G\rightarrow GL_{n}(k)$ if and only if $\rho^{*}(\gamma)=0$ in $H^{2}(G;k^{*})$ for the restriction map $\rho^{*}:H^{2}(PGL_{n}(k);k^{*}) \rightarrow H^{2}(G;k^{*})$
I have not been able to do much other than unwinding the definitions. Kindly provide a solution.

Consider any homomorphism $\rho: G \rightarrow PGL_n(k)$. Let $p_0: PGL_n(k) \rightarrow GL_n(k)$ be any set-theoretic lift, $p=p_0 \circ \rho$ is a set-theoretic lift of $\rho$. $\rho$ lifts iff there is some map $t: G \rightarrow k^{\times}$ such that $p/t$ is a group homomorphism.
$p/t$ is a group homomorphism iff for all $g,h \in G$, $p(g)^{-1}p(gh)p(h)^{-1}=t(g)^{-1}t(gh)t(h)^{-1}$.
Now, define $p’: (g,h) \in G^2 \longmapsto p(g)^{-1}p(gh)p(h)^{-1} \in k^{\times}$, $p_0’: (g,h)\in PGL_n(k)^2 \longmapsto p_0(g)^{-1}p_0(gh)p_0(h)^{-1} \in k^{\times}$, so that $p’=p_0’ \circ \rho^{\times 2}$.
Then it’s enough to show that:
1: $p’$ (and in particular $p’_0$ for $\rho$ being the identity) is a $2$-cocycle.
2: $p’$ is a coboundary iff $\rho$ lifts.
The second part follows from simply unwinding the definition of coboundary given the considerations above (then $p’=-dt$).
The first part is a computation using the fact that the image of $p’$ is central in $GL_n(k)$. We have \begin{align*}
dp’(g,h,k)&=p’(h,k)p’(gh,k)^{-1}p’(g,hk)p’(g,h)^{-1}\\
&=p(h)^{-1}p(hk)p(k)^{-1}p(k)p(ghk)^{-1}p(gh)p’(g,hk)p’(g,h)^{-1}\\
&=p(h)^{-1}p’(g,hk)p(hk)p(ghk)^{-1}p(gh)p’(g,h)^{-1}\\
&=p(h)^{-1}p(g)^{-1}p(ghk)p(hk)^{-1}p(hk)p(ghk)^{-1}p(gh)p’(g,h)^{-1}\\
&=p(h)^{-1}p(g)^{-1}p(gh)p’(g,h)^{-1}\\
&=p(h)^{-1}p’(g,h)p(h)p’(g,h)^{-1}\\
&=1.
\end{align*}
