  The $2$-primary Hopf invariant $1$ elements in the stable homotopy groups of
spheres form the most accessible family of elements. In this paper we explore
some properties of the $\mathcal{E}_\infty$ ring spectra obtained from certain
iterated mapping cones by applying the free algebra functor. In fact, these are
equivalent to Thom spectra over infinite loop spaces related to the classifying
spaces $B\mathrm{SO},\,B\mathrm{Spin},\,B\mathrm{String}$. We show that the
homology of these Thom spectra are all extended comodule algebras of the form
$\mathcal{A}_*\square_{\mathcal{A}(r)_*}P_*$ over the dual Steenrod algebra
$\mathcal{A}_*$ with $\mathcal{A}_*\square_{\mathcal{A}(r)_*}\mathbb{F}_2$ as
an algebra retract. This suggests that these spectra might be wedges of module
spectra over the ring spectra $H\mathbb{Z}$, $k\mathrm{O}$ or $\mathrm{tmf}$,
however apart from the first case, we have no concrete results on this.
