  We show the existence of a $2$-parameter family of properly
Alexandrov-embedded surfaces with constant mean curvature $0<H\leq\frac{1}{2}$
in $\mathbb{H}^2\times\mathbb{R}$. They are symmetric with respect to a
horizontal slice and a $k$ vertical planes disposed symmetrically, and extend
the so called minimal saddle towers and $k$-noids. We show that the orientation
plays a fundamental role when the mean curvature is positive via the sister
conjugate minimal surface from $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ or
$\mathrm{Nil}_3$. We also discover new complete examples that we call
$(H,k)$-nodoids, whose $k$ ends are asymptotic to vertical cylinders from the
convex side, often giving rise to non-embedded examples. In the discussion of
embeddedness, we prove that the Krust property does not hold for any $H>0$,
i.e., there are minimal graphs over convex domains in
$\widetilde{\mathrm{SL}}_2(\mathbb{R})$, $\mathrm{Nil}_3$ or the Berger
spheres, whose sister conjugate surfaces with constant mean curvature $H$ in
$\mathbb{H}^2\times\mathbb{R}$ are not graphs.
