In his 1973 paper "The nilpotency of elements of the stable homotopygroups of spheres", Goro Nishida calculates, among other things, the homology of extended powers. Letting $M_k = S^k \cup_2 D^{k+1}$ be the Moore space in dimension $k$ of the cyclic group $\mathbb{Z}/(2)$, denote by $D_n(M_k) = (E\Sigma_n)_+\wedge (M_k)^{\wedge n}$ the $n$-th extended power of $M_k$. On the one hand, he claims that in the range $0<i<n/2$ one has isomorphisms $$ H_{nk+i}(D_n(M_k);\mathbb{Z}/(2))\cong \mathcal{A}_2^i, $$ as $\mathbb{Z}/(2)$-modules to the mod-2 Steenrod algebra.
On the other hand, the mod-2 homology of $D_n(M_k)$ has a basis given through Dyer-Lashof operations. Letting $x,y$ with $|x|=k,|y|=k+1$ denote a basis of $\tilde H_*(M_k;\mathbb{Z}/(2))$, it is a well-known result that for $Q(M_k)$ the infinite loopspace associated to $M_k$, the following holds:
As a commutative, graded algebra, $H_*(Q(M_k);\mathbb{Z}/2)$ is generated by iterated Dyer-Lashof operations applied to $x$ and $y$, i.e. elements of the form $Q^I(x)$ and $Q^J(y)$ with $I,J$ admissible and excess $e(I)>k,e(J)>k+1$.
The homology of $D_n(M_k)$ then corresponds to the submodule $A(n)_*\subseteq H_*(Q(M_k);\mathbb{Z}/(2))$ generated by the monomials of height $n$.
Here my issue arises. Picking (random) values, $n=10, k=5, i=2$ yields $$ A(10)_{52} = A(n)_{nk+i} \cong H_{nk + i}(D_n(M_k);\mathbb{Z}/(2)) \cong \mathcal{A}_2^i= \mathcal{A}_2^2 , $$ The LHS has rank 4, generators being given by $x^{8}y^2, x^8Q^7x, x^6Q^6xQ^6x, x^7yQ^6x$. The RHS has rank 1, a generator is given by $\text{Sq}^2$.
I would like to know which of the two paths to calculating the homology of $D_n(M_k)$ is wrong, or if I am overlooking something simple.
Also let me know if I should post this on MathOverflow.