Permutative APS

Definition
A permutative APS of groups is an APS of groups $$(G,\Phi)$$ equipped with an APS-on-APS action on it by the permutation IAPS, having the following property:

If $$\sigma: \{ 1,2,3,\dots,m+n \} \to \{ 1,2,3,\dots,m+n \}$$ is a permutation defined as:

$$\sigma(j) = n + j, \qquad 1 \le j \le m$$

and:

$$\sigma(j) = j - m, \qquad m + 1 \le j \le n$$

Then, for $$g \in G_m, h \in G_n$$, we have:

$$\Phi_{n,m}(h,g) = \sigma \cdot \Phi_{m,n}(g,h)$$