Let $Y$ be a H-space and suppose $X$ is a pointed retract of $Y$ with continuous pointed maps $s,r$.
My thinking so far;
If Y is a H-space then there is a map $m:Y$x$Y \rightarrow Y$ such that $m$ composed with the inclusion map is homotopic to the identity. So we need an $n$ that does the same but for maps on $X$. So if $n=r \circ m \circ s$x$s$ then I have a map that satisfies the condition for a H-space so I am done? Or do i need to somehow show that this composed with the inclusion map is the identity on $X$?