In the Fomenko's Book,
we find the characterization of stable cohomology operations $O^S(q,G,H)$ as the projective limit of $$\cdots \longrightarrow \mathcal{H}^{q+n+1}(K(G,n+1);H)\longrightarrow\mathcal{H}^{q+n}(K(G,n);H)\longrightarrow \cdots\longrightarrow \mathcal{H}^{q+1}(K(G,1);H) $$ where the maps $f_n$ are the composition $$\mathcal{H}^{n+q}(K(G,n);H)\xrightarrow{\quad i_n^*\quad} \mathcal{H}^{n+q}(\Sigma K(G,n-1);H) $$ $$\xrightarrow{\Sigma^{-1}} \mathcal{H}^{n+q-1}(K(G,n-1);H) .$$
The map $\iota_n$ is explained here: Suspension map and Stable cohomology operation as inverse limit.
In the case $G=H=\mathbb{Z}_2$, by Freudenthal theorem and Hurewicz theorem, we obtain a range where $f_n$'s are isomorphisms, so we deduce that the projective limit is a finite limit and $$O^S(q,\mathbb{Z}_2,\mathbb{Z}_2)\simeq \mathcal{H}^{n}(K(\mathbb{Z}_2,q);\mathbb{Z}_2).$$
Now, I tried to verify that the maps are isomorphisms for $n$ large, but I obtain the inverse condition, $q<n<2q$.
Where is the mistake?
Secondly, by Serre theorem, that it proved $$\mathcal{H}^{*}(K(\mathbb{Z}_2,q);\mathbb{Z}_2)\simeq \mathbb{Z}_2[Sq^I(e_q)]$$ (where $I$ is admissible with excess less than $q$ and $e_q$ is a fundamental class of $\mathcal{H}^{q}(K(\mathbb{Z}_2,q);\mathbb{Z}_2)$) we obtain as corollary, that the algebra generated by Steenrod squares is isomorphic to $\tilde{\mathcal{H}}^{n}(K(\mathbb{Z}_2,q);\mathbb{Z}_2)$ for $q\le n\le 2n$ (and the isomorphism is graded).
Now, my questions are: first of all which is the mistake in projective limit and then, why, by the corollary of Serre, we obtain an isomorphism between the algebra generated by Steenrod squares and $O^S(q,\mathbb{Z}_2,\mathbb{Z}_2)$.
Thanks you in advance.