A generalization of the anti-automorphisms of a group Recently, there was a question about anti-automorphisms of a group $G$.  In this case, the inversion map $\iota : G \rightarrow G : x \mapsto x^{-1}$ is an anti-automorphism and $\iota \mathrm{Aut}(G)$ is the coset of all anti-automorphisms.
Let $K$ be a permutation group of degree $n$ and let $\pi \in K$.  A natural generalization is a $\pi$-automorphism of the group $G$, where we say that a bijection $\phi : G \rightarrow G$ is a $\pi$-automorphism if for all $x_1, \ldots, x_n \in G$,
$$\phi(x_1 \cdots x_n) = \phi(x_{\pi(1)}) \cdots \phi(x_{\pi(n)})$$
We'll say that $\phi$ is a $K$-automorphism if it is a $\pi$-automorphism for some $\pi \in K$.
Now it's clear that the $K$-automorphisms of $G$ form a group and it's not hard to tell what the cosets of $\mathrm{Aut}(G)$ are in this group.
What I'm wondering if there are examples of groups that have interesting $K$-automorphisms or if the structure of the $K$-automorphisms is understood.

An example of a $\langle (1\,2\,3)\rangle$-automorphism is $f(x)=ex$, where $e$ is some central involution.
