Let $(X, x_0)$ be a based topological space, and $\Omega X$ its based loop space. The group of path components of $\Omega X$ is $\pi_0(\Omega X) = \pi_1(X, x_0)$. For brevity, let's call this group $G$. Since $\Omega X$ is a loop space, all of its components are homotopy equivalent, and so if we define $\Omega_1 X$ to be the component associated to $1 \in G$ (i.e., the null-homotopic loops), then there is a homotopy equivalence

$$\varphi: G \times \Omega_1 X \to \Omega X.$$

Explicitly, $\varphi(g, f) = r(g)*f$, where $r: G \to \Omega X$ is a section of $\pi_0$, and $*$ is concatenation of loops. My question is about the extent to which this map is a loop map (or $A_\infty$ map).

We can equip the domain of $\varphi$ with the structure of an $H$-space with multiplication

$$(g_1, f_1) \cdot (g_2, f_2) := (g_1 g_2, r(g_2)^{-1}*f_1 * r(g_2) * f_2);$$

i.e., a sort of semidirect product structure where $G$ acts on $\Omega_1 X$ by conjugation of loops (via $r$). Then $\varphi$ is an $H$-map with respect to this multiplication: i.e., it is multiplicative up to homotopy. However, the domain is not a strictly associative monoid with respect to the multiplication $\cdot$, nor is $\varphi$ a strict morphism.

So, my question is: is there an $A_\infty$ structure on $G \times \Omega_1 X$ agreeing the multiplicative structure described above (up to homotopy) and a map $\Phi: G \times \Omega_1 X \to \Omega X$, homotopic $\varphi$, which is an $A_\infty$ map?