Is there a criterion for a based map $f:G\to H$ between deloopable pointed spaces to be deloopable, i.e., that there is a based map $g:X\to Y$ between path-connected pointed spaces such that $G\simeq \Omega X$, $H\simeq \Omega Y$ and $f=\Omega g$?

You may assume that you know $G$ and $H$ are already deloopable, i.e., are $A_\infty$-spaces such that the induced monoid on $\pi_0$ is a group (this has been discussed before, see for example this question.)