The problem of determining spherical classes in \(H_\ast (X)\) is not always easy, e.g., in the case of \(X=Q\mathbb{S}^0=\mbox{colim}\,\Omega^k\mathbb{S}^k\) it is an open problem as one can see in e.g., [\textit{E. B. Curtis}, Ill. J. Math. 19, 231--246 (1975; Zbl 0311.55007)]. The problem of determining spherical classes in finite loop spaces \(\Omega^k\mathbb{S}^{m+k}\) is also open, although some progress for small values of \(l\) has been made by the author in [Topology Appl. 224, 1--18 (2017; Zbl 1369.55007)]. The author follows the philosophy that, at least on the level of algebra, the Hurewicz and Boardman homomorphisms are dual and sometimes the dual problem might be easier to tackle. For a nice topological space \(X\), working at the prime \(p=2\), the author considers the ``unstable Boardman map'' (homomorphism if \(k>0\)) \[b : [X,\Omega^k\mathbb{S}^{m+k}]\to \mbox{Hom}_{\mathbb{Z}/2}(H^\ast(\Omega^k\mathbb{S}^{m+k}),H^\ast(X))\] defined by \(b(f)=f^\ast\), where \(k\ge 0\) and \(m\ge 0\). Classic maps, such as the Kahn-Priddy map, are used to provide examples of \(X\) so that the map \(b\) is nonzero in many dimensions. \par Some generalities of the above for \(E\)-cohomology with a nice ring spectrum \(E\) are investigated as well.