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We describe the relation between G-character varieties, $X_G(\Gamma)$, and$G/H$-character varieties, where $H$ is a finite, central subgroup of $G.$ Inparticular, we find finite generating sets of coordinate rings$C[X_{G/H}(\Gamma)]$ for classical groups $G$ and $H$ as above. Using thisapproach we find an explicit description of $C[X_{SO(4,C)}(F_2)]$ for the freegroup on two generators, $F_2.$  In the second part of the paper, we prove several properties ofSO(2n,C)-character varieties. This is a particularly interesting class ofcharacter varieties because unlike for all other classical groups G, thecoordinate rings $C[X_{G}(\Gamma)]$ are generally not generated by tracefunctions $\tau_\gamma$, for $\gamma\in \Gamma$, for G=SO(2n,C). In fact, weprove that the coordinate ring $C[X_{SO(2n,C)}(\Gamma)]$ is not even generatedby "generalized trace functions," $\tau_{\gamma,V},$ for all $\gamma\in \Gamma$and all representations $V$ of $SO(2n,C)$ for $n=2$ and groups $\Gamma$ ofcorank $\geq 2$.

G-Character varieties for G=SO(n,C) and other not simply connected groups