  Given a spectral triple on a unital $C^{*}$-algebra $A$ and an equicontinuous
action of a discrete group $G$ on $A$, a spectral triple on the reduced crossed
product $C^{*}$-algebra $A\rtimes_r G$ was constructed by Hawkins, Skalski,
White and Zacharias in [On spectral triples on crossed products arising from
equicontinuous actions, Math. Scand. 113(2) (2013) 262-291], extending the
construction by Belissard, Marcolli and Reihani in [Dynamical systems on
spectral metric spaces, preprint (2010), arXiv:1008.4617], by using the
Kasparov product to make an ansatz for the Dirac operator. Supposing that the
triple on $A$ is equivariant for an action of $G$, we show that the triple on
$A\rtimes_r G$ is equivariant for the dual coaction of $G$. If moreover an
equivariant real structure $J$ is given for the triple on $A$, we give
constructions for two inequivalent real structures on the triple $A\rtimes_rG$.
We compute the KO-dimension with respect to each real structure in terms of the
KO-dimension of $J$ and show that the first and the second order conditions are
preserved. Lastly, we characterise an equivariant orientation cycle on the
triple on $A\rtimes_rG$ coming from an equivariant orientation cycle on the
triple on $A$. We show, along the paper, that our constructions generalize the
respective constructions of the equivariant spectral triple on the
noncommutative $2$-torus.
