We characterize the obstructions to the Erdős-Pósa property of $A$-paths in unoriented group-labelled graphs. As a result, we prove that for every finite abelian group $Γ$ and for every subset $Λ$ of $Γ$, the family of $Γ$-labelled $A$-paths whose lengths are in $Λ$ satisfies the half-integral Erdős-Pósa property. Moreover, we give a characterization of such $Γ$ and $Λ\subseteqΓ$ for which the same family of $A$-paths satisfies the full Erdős-Pósa property.