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Let $G$ be the finite simple Chevalley group of type $^2E_6(2)$. It has aSchur multiplier of type $C_2^2 \times C_3$. We determine the ordinarycharacter tables of the central extensions $3.G$, $6.G$, $(2^2\times 3).G$ of$G$ and their extensions by an automorphism of order $2$, that is $3.G.2$,$6.G.2$ and $(2^2\times 3).G.2$.  Furthermore we determine all Brauer trees of all groups of type $Z.G.A$(where $Z$ is central in $Z.G \lhd Z.G.A$ and $A \cong Z.G.A/Z.G$) for whichthe ordinary character table is known.

Characters and Brauer trees of the covering group of $^2E_6(2)$