  Hyperoctahedral homology for involutive algebras is the homology theory
associated to the hyperoctahedral crossed simplicial group. It is related to
equivariant stable homotopy theory via the homology of equivariant infinite
loop spaces. In this paper we show that there is an E-infinity algebra
structure on the simplicial module that computes hyperoctahedral homology. We
deduce that hyperoctahedral homology admits Dyer-Lashof homology operations.
Furthermore, there is a Pontryagin product which gives hyperoctahedral homology
the structure of an associative, graded-commutative algebra. We also give an
explicit description of hyperoctahedral homology in degree zero. Combining this
description and the Pontryagin product we show that hyperoctahedral homology
fails to preserve Morita equivalence.
