  We consider the `unstable Boardman map' (homomorphism if $k>0$)
$$b:\pi^{m+k}\Sigma^k\Omega^lS^{n+l}\simeq[\Omega^lS^{n+l},\Omega^kS^{m+k}]\longrightarrow
\mathrm{Hom}(H_*\Omega^lS^{n+l},H_*\Omega^kS^{m+k})$$ defined by $h(f)=f_*$. We
work at the prime $2$, with $k=0$, and determine the image for various in the
following cases : (1) $m=n$ and $l>0$ arbitrary; (2) $m>n$ and $l=1$. We
observe that in most of the cases the image is trivial with the exceptions
corresponding to the cases when either there is a (commutative) $H$-space
structure on $S^n$ or there is a Hopf invariant one element.
