A generalized Kervaire Problem in stable homotopy groups of spheres

We will give an elementary self-contained description of the Mahowald element in the stable homotopy group of spheres ${\mathrm{\Pi }}_{2^l}$, $l\ge 3$. Using this construction, we prove that a generalized Kervaire Problem, formulated by the author in \cite{A} is solved positively.