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We show the existence of a $2$-parameter family of properlyAlexandrov-embedded surfaces with constant mean curvature $0\leqH\leq\frac{1}{2}$ in ${\mathbb{H}^2\times\mathbb{R}}$. They are symmetric withrespect 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 theorientation plays a fundamental role when $H>0$ by analyzing their conjugateminimal 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 overcurves of geodesic curvature $2H$ from the convex side, often giving rise tonon-embedded examples if $H>0$.  In the discussion of embeddedness of the constructed examples, we prove thatthe Krust property does not hold for any $H>0$, i.e., there are minimal graphsover convex domains in $\widetilde{\mathrm{SL}}_2(\mathbb{R})$,$\mathrm{Nil}_3$ or the Berger spheres, whose conjugate surfaces with constantmean curvature $H$ in $\mathbb{H}^2\times\mathbb{R}$ are not graphs.

A construction of constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$ and the Krust property