It is a theorem that a finite nontrivial group $G$ has no proper nontrivial characteristic subgroups if and only if $G \cong S^n$ where $S$ is simple and $n > 0$ is the number of copies of $S$ in a direct product. Call an automorphism $\varphi \in \operatorname{Aut}(G)$ *irreducible* if $\varphi(H) \not\subseteq H$ for all proper nontrivial **normal** $H \subseteq G$. (I am just coining this term, please let me know if there is an actual name for this.) When $S$ is abelian, irreducible automorphisms are precisely irreps of the group $\mathbb{Z}$ over the vector space $\mathbb{F}_p^n$ and are precisely characterized by irreducible polynomials in the ring $\mathbb{F}_p[t]$ of degree $n$, excluding the polynomial $t$ because it corresponds to the $0$ map. What is the situation like when $S$ is non-abelian?

Edit: bolded text added.

normalsubgroups. $\endgroup$