  We provide families of compact astheno-K\"ahler nilmanifolds and we study the
behaviour of the complex blowup of such manifolds. We prove that the existence
of an astheno-K\"ahler metric satisfying an extra differential condition is not
preserved by blowup. We also study the interplay between Strong K\"ahler with
torsion metrics and geometrically Bott-Chern metrics. We show that
Fino-Parton-Salamon nilmanifolds are geometrically-Bott-Chern-formal, whereas
we obtain negative results on the product of two copies of primary Kodaira
surface, Inoue surface of type $\mathcal{S}_M$ and on the product of a Kodaira
surface with an Inoue surface.
