Lifting to mod2 Moore spaces

We try to investigate whether an element of order 2 in homotopy groups of spheres has a lift to an element of homotopy groups of mod2 Moore spaces or not. A typical element of the affirmative property is Thomeier's element of pi(8n+7)(S4n+3). And the Whitehead square [l(2n+1), l(2n+1)] Of the identity class l(2n+1) of S2n+1 is a negative example except for n = 0, 1 and 3.