$n$-inner automorphisms of finite groups

We refer to an automorphism $g$ of a group $G$ as $n$-inner if given any subset $S$ of $G$ with cardinality less than $n$, there exists an inner automorphism of $G$ agreeing with $g$ on $S$. Hence $g$ is 2-inner if it sends every element of $G$ to a conjugate. New examples are given of outer $n$-inner automorphisms of finite groups for all natural numbers $n\geq 2$.