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.

I was interested to this group-theoretical property for this very reason. I was studying the class of hereditarily connected quandles (I called them superconnected quandles in a couple of paper I wrote).

For the results you are referring to, do you need to assume that $G=\langle x^G\rangle$?

I see. Maybe I also need to assume also that $x^G\cap C_G(y)=\{y\}$ for every $y\in x^G$.

Thanks for your reply. In the first part of your argument you are using Burnside's p-complement theorem, right? What if I assume that <x> has a normal complement and $C_G(x)=N_G(x)$? Is there a way to prove that $G$ has the desired property?

Thanks for your reply. Are the properties i) and ii) you mentioned at the end of your post equivalent to the one I am interested in, or they are sufficient? I think in ii) you mean that $[\langle x ,x^g\rangle,x]=\{ [h,x], h\in \langle x, x^g\rangle\}$, right?

That was easy! Thanks.

I see, so I need to consider the aperiodic finite index subgroup to conclude. Is there any reference where I can find all of it? Thanks.

Thank you for your reply. So for sure I can find an infinite simple groups with exponent a power of 2, as the simple factor of the Burnside group B(2,2^m) for m large enough, right? Do you have any good reference for this? Thanks again.