Rational homotopy type and computability

Given a simplicial pair $(X,A)$, a simplicial complex $Y$, and a map $f:A \to Y$, does $f$ have an extension to $X$? We show that for a fixed $Y$, this question is algorithmically decidable for all $X$, $A$, and $f$ if and only if $Y$ has the rational homotopy type of an H-space. As a corollary, many questions related to bundle structures over a finite complex are likely decidable.