Regular orbits for automorphisms of finite simple groups

Let $G$ be a finite group and $f$ be an automorphism of $G$. We say that $f$ has a regular orbit if there exists $x\in G$ such that $|x^f|=|f|$. If $G$ is abelian it is known that every automorphism of $G$ has a regular orbit.

If $L$ is a finite simple non-abelian group, can I always find an automorphism of $L$ with no regular orbit?

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Edit: I was actually looking for an example of a finite non abelian simple group 𝐿 with an automorphism with no regular orbit. Sorry for the unclear question.