Decidability of the extension problem for maps into odd-dimensional spheres

In a recent paper, it was shown that the problem of existence of a continuous map $X\to Y$ extending a given map $A\to Y$ defined on a subspace $A \subseteq X$isundecidable,evenfor$Y$ an even-dimensional sphere. In the present paper, we prove that the same problem for $Y$ an odd-dimensional sphere is decidable. More generally, the same holds for any $d$-connected target space $Y$ whose homotopy groups ${\pi }_{k}Y$ are finite for $k>2d$.