A representative $\alpha$ of a cohomology class $[\alpha]\in H^n(X;\pi_n(X))$ is equivalent to a map from $X$ into an Eilenberg-Mac Lane space as $\alpha:X\to K(\pi_n(X),n)$. After applying a covariant functor $[S^n,\_]$, we get a homomorphism between homotopy groups: $\alpha_*:\pi_n(X)\to \pi_n(X)$.

Then, what is the condition on $\alpha$ so that $\alpha_*$ becomes an isomorphism? Rather vaguely, is there any way of describing $\alpha_*$ in terms of $\alpha$ or the other way around?

2nd edit:

Sorry for missing the motivation but appreciate the answer. I was wondering if there is any relationship between some algebraic structure in $H^n(X;\pi_n(X))$ and the property that the induced homomorphism is an isomorphism, something like "a generator induces an isomorphism". Is this true or does it make any sense to ask at all? And I also wonder if there is any general argument on what does algebraic operations in the cohomology do on induced homomorphisms. BTW, I also need this for the case when $n=1$.

3rd edit:

Thanks for the comments. I guess these previous questions were sort of ill-stated by being over simplified and generalized. The starting point was this:

First, $[BSO,K(\pi_1(O),2)]\cong H^2(BSO;\pi_1(O))$ and the cohomology is generated by the second Stiefel-Whitney class $w_2$. Then I guess that $(w_2)_*:\pi_2(BSO)\to \pi_1(O)$ should be an isomorphism but I can't be sure of that.

The same line of thought applies to $[B{\rm Spin},K(\pi_3(SO),4)]\cong H^4(B{\rm Spin};\pi_3(SO))$ and the latter cohomology is generated by the half of first Pontrjagin class $\frac{1}{2}p_1$. I also guess that this class would induce the isomorphism $(\frac{1}{2}p_1)_*:\pi_4(B{\rm Spin})\to \pi_3({\rm SO})$ but I'm not sure too.

What I was wondering was that, if $w_2$ and $\frac{1}{2}p_1$ do induce isomorphisms, then do they so because of the algebraic property of being generators (instead of using seemingly ad hoc constructions and special properties of theses spaces) so that this would generalized easily into other cases.