 Solution

Use the axes actually printed: the two hydrogens lie in the $xz$ plane, $z$ is the principal twofold axis, and $y$ is perpendicular to that plane. For a Cartesian-displacement group representation, only atoms left at their own sites contribute to the trace; their vector components must also be transformed. In the order $(E,C_2(z),\sigma (xz),\sigma (yz))$, identity fixes six components, the twofold rotation exchanges the hydrogens, the $xz$ reflection fixes both hydrogens with vector matrix $\mathrm{diag}(1,-1,1)$, and the $yz$ reflection exchanges them. ThereforeThe nonzero mirror character of a representation is $2(1-1+1)=2$, not six.