Let $G$ be a group and assume $G$ finite for simplicity. Let $k$ be a field of characteristic $0$. Let $\rho: G \to \operatorname{GL}_n(k)$ be a representation of $G$. To $\rho$ we can associate another representation $\operatorname{ad}(\rho): G \to \operatorname{GL}(\operatorname{M}_n(k))$ defined by $g \mapsto (A \mapsto \rho(g)A\rho(g)^{-1})$.
How much does $\operatorname{ad}(\rho)$ know about $\rho$? Clearly, if $\rho(g)$ lies in the centre $Z$ of $\operatorname{GL}_n(k)$ then $\operatorname{ad}(\rho)(g) = 1$. Can we recover $\overline \rho: G \to \operatorname{GL}_n(k)/Z$ from $\operatorname{ad}(\rho)$?
We can assume that $k$ is algebraically closed, if it makes a difference. Apologies if this is a well-known result.