For space pairs $(X\_1,A\_1)$ and $(X\_2,A\_2)$ ,is there a groups M,such that $\pi\_{k}(X\_1\vee X\_2,A\_1\vee A\_2)\cong \pi\_k(X\_1,A\_1)\oplus M$?



is $\pi_k(X_1,A_1)$ a direct summand of $\pi_{k}(X_1\vee X_2,A_1\vee A_2)$
The pair $(X\_1,A\_1)$ is a retract of the pair $(X\_1\vee X\_2,A\_1\vee A\_2)$, so the map $\pi\_k(X\_1,A\_1)\to \pi\_k(X\_1\vee X\_2,A\_1\vee A\_2)$ is a split injection for all $k$. If $k\ge 3$ then the groups are abelian, and your result follows with $M$ the cokernel. 


