One more contradiction in mathematics. I hope someone corrects me in the following statements. I will mark them by ?, so if anything simply not true please leave a comment
Let X be a topological space with Steenrod-coaction $$\psi: H_\bullet(X)\rightarrow \mathcal{A}\otimes H_\bullet(X)$$ (set coefficients to $\mathbb{F}_2$ by default, put $\mathcal{A}$=$\mathcal{A}_2$).
Then, the image of Hurewicz map $$\Gamma:\pi_\bullet(X)\rightarrow H_\bullet(X)$$ consists of $\psi$-primitive elements in $H_\bullet(X)$ ?, so $\psi(\Gamma(s))=1\otimes s$ for any $s\in \pi_\bullet(X)$ -- it is clear from functoriality of coaction applied to $$s:S^n\to X$$ (the same argument can be applied to any multiplicative homology-theory).
On the other hand we have comultiplication on $H_\bullet(X)$ given by diagonal $$\Delta:X\to X\times X$$ and it maps any spheroid $$\Gamma(s)\in H_n(X)$$ to ? $$1\otimes \Gamma(s) + \Gamma(s)\otimes 1$$ this follows from restriction to the case $S^n\rightarrow S^n\times S^n$.
So spheroids in $H_\bullet(X)$ are primitive with respect to comultiplication $\Delta$ and to coaction by $\mathcal{A}$ ?.
Now I want to check consistency of this conclusion by inspecting concrete example. Take $$X=BU(\infty,\infty)$$ be a H-space, then $H_\bullet(X,\mathbb{Z})$ is a Hopf-algebra isomorphic to ?: $$H_\bullet(X,\mathbb{Z})\sim \mathbb{Z}[b_i|i\geq 0, b_0=1]$$ with clear multiplication $\mu$ and comultiplication $\Delta$: $$\Delta b_\bullet = b_\bullet\otimes b_\bullet$$ given by grading-parts of both sides ($b_\bullet$ are coming from $H_\bullet(\mathbb{C}P^\infty)=\mathbb{Z}\langle b_i\rangle$ and classifying map $\mathbb{C}P^\infty\rightarrow BU(\infty)$)
Steenrod-coaction on $\mathbb{C}P^\infty$ and consequently on $X$ given by: $$\psi: b_\bullet\rightarrow \sum\limits_{i\geq 0} b_i [\xi_\bullet^2]^i$$ where $$[\xi_\bullet^2]=\sum\limits_{j\geq 0} \xi_j^2$$ equal to sum of squares of Milnor-generators of $\mathcal{A}_2$ (deg$\xi_i=2^i-1$, deg$b_i=2i$)
From Bott-periodicity $\pi_\bullet(X)=\mathbb{Z}[t]$, deg t=2 and Hurewicz injectivity over $\mathbb{Q}$ using Milnor-Moore theorem we can identify Hurewicz-image with $\Delta$-primitive elements in $H_\bullet(X)$. Following Adams blue-book there is complete description of $\Delta$-primitive elements in coalgebra $H_\bullet(X,\mathbb{Z})$, for example one can check by hand that:
$b_1$ in degree 1
$b_1^2-2b_2$ if degree 2
$b_1^3-3b_1b_2+3b_3$ in degree 3
and so on are $\Delta$-primitive (for general case left side is given by expression of Newton-polynomials through elementary-symmetrical)
In that way we see for example that ? $$\Gamma(\pi_3(X))=\mathbb{Z}\langle b_1^3-3b_1b_2+3b_3\rangle\subset H_3(X,\mathbb{Z})$$ So, after reduction mod 2 this element should be $\mathcal{A}_2$-primitive, but it does not: $$\psi(b_1^3-3b_1b_2+3b_3)=1\otimes (b_1^3-3b_1b_2+3b_3) + \xi_1^2\otimes b_1^2\mod 2$$ because $b_1,b_3$ are $\psi$-primitive and $\psi(b_2)=1\otimes b_2 + \xi_1^2\otimes b_1$