Let $\mathcal{H}(f)$ be the Hopf invariant of a map $f:\mathbb{S}^{4n-1}\to \mathbb{S}^{2n}$.

When is the suspension map $$ \sigma:\{f\in \pi_{4n-1}(\mathbb{S}^{2n}):\, \mathcal{H}(f)\neq 0\}\to \pi_{4n}(\mathbb{S}^{2n+1}) $$ a non-zero map? When is it a surjection?

Clearly it is a surjection when $n=1$, but I am interested in the higher dimensional cases. In other words, I would like to know if there exist non-trivial elements in $ \pi_{4n}(\mathbb{S}^{2n+1})$, $n\geq 2$, that are suspensions of a map with a non-zero Hopf invariant.