The ($ \bmod 2$) Steenrod Algebra $\cal A$ is a quotient of the universal Steenrod algebra $Q$, the homogeneous quadratic algebra of cohomology operations in the category of $H_\infty$-ring spectra.
We consider a quadratic linear algebra $Q_1$
obtained by suitably changing the generators of $Q$ in order to make the quotient $ \pi: Q \rightarrow \cal A $ a map of augmented algebras.
 It turns out that the map induced on Hochschild cohomology is a monomorphism. We also compute $ \Ext_{Q_1} (\B F_2, \B F_2)$ under an hypothesis of koszulness on $Q$. 

The $E_1$-term of the Adams spectral sequence embeds in $Q$. Here, the differential $d_1$ can be described in a very simple way. This fact suggests that   $\Ext_{Q_1}(\B F_2, \B F_2)$ could be the right object to investigate in order to get a combinatorial description of $d_2$.