Finite simple non-abelian group not implies every element is automorphic to its inverse

Statement
It is possible to have a fact about::finite simple non-abelian group $$G$$ (i.e., a fact about::simple non-abelian group that is also finite) that is not a fact about::group in which every element is automorphic to its inverse. In other words, there exists $$g \in G$$ such that there is no automorphism of $$G$$ sending $$g$$ to $$g^{-1}$$.

Example of the Mathieu group
The Mathieu group $$M_{11}$$ is a complete simple non-abelian group, viewed as a subgroup of the alternating group of degree eleven. In particular, every automorphism of $$M_{11}$$ is inner. However, $$M_{11}$$ contains an $$11$$-cycle, which is a cycle of odd length that is $$3$$ mod $$4$$. This is not conjugate to its inverse even in the alternating group of degree eleven, which contains $$M_{11}$$. Hence, there is an element that is not conjugate to its inverse. Since every automorphism is inner, this completes our example.