  We show the existence of a $2$-parameter family of properly
Alexandrov-embedded surfaces with constant mean curvature $0\leq
H\leq\frac{1}{2}$ 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 $H>0$ by analyzing their conjugate
minimal surfaces in $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ or
$\mathrm{Nil}_3$. We also discover new complete examples that we call
$(H,k)$-nodoids, whose $k$ ends are asymptotic to vertical cylinders over
curves of geodesic curvature $2H$ from the convex side, often giving rise to
non-embedded examples if $H>0$.
  In the discussion of embeddedness of the constructed examples, 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 conjugate surfaces with constant
mean curvature $H$ in $\mathbb{H}^2\times\mathbb{R}$ are not graphs.
