  In this largely expository paper we extend properties of the homological
duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the
Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke
algebra of a finite central extension of a reductive $p$-adic group. The most
important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is
concentrated in a single degree for irreducible representations and that it
gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like
property). Along the way we also study Grothendieck--Serre duality with respect
to the Bernstein center and provide a proof of the folklore result that on
admissible modules this functor is nothing but the contragredient duality. We
single out a necessary and sufficient condition for when these three dualities
agree on finite length modules in a given block. In particular, we show this is
the case for all cuspidal blocks as well as, due to a result of Roche, on all
blocks with trivial stabilizer in the relative Weyl group.
