Permutation IAPS

Symbol-free definition
The permutation IAPS is an IAPS of groups where the $$n^{th}$$ member is the symmetric group $$S_n$$, and where the block concatenation map $$S_m \times S_n \to S_{m+n}$$ is defined as the permutation that permutes the first $$m$$ symbols according to the permutation in $$S_m$$ and the next $$n$$ symbols according to the permutation in $$S_n$$.

Definition with symbols
The permutation IAPS is an IAPS of groups where the $$n^{th}$$ member is $$S_n$$ and the block concatenation map $$\Phi_{m,n}: S_m \times S_n \to S_{m+n}$$ is defined as follows:

Given a permutation $$g \in S_m$$ and a permutation $$h \in S_n$$, the permutation $$\Phi_{m,n}(g,h)$$ is defined as the following permutation on $$\{1,2,3,\dots,m+n\}$$. It sends $$i \in \{ 1,2,3,\dots,m \}$$ to $$g(i)$$, and sends $$j \in \{m + 1, m+2, \dots,m+n\}$$ to $$h(j-m) + m$$.

Examples
As an example, consider $$m = 3$$, $$n = 4$$. Let $$g = (1,3)$$, and $$h = (1,3,4)$$. Then:

$$\Phi_{3,4}(g,h) = (1,3)(4,6,7)$$