I'm supervising an undergraduate research project. Among other things, I've got the student to look at this paper of Gene Kopp and John Wiltshire-Gordon. This question arose from a missing complex conjugate in something the student wrote.

Let $g$ be an element of a finite group $G$, and $w$ a word in $n$ variables. If you evaluate the word on all $n$-tuples of elements of $G$, does it give $g$ and $g^{-1}$ the same number of times?

I thought the answer must be "no", but found it frustratingly difficult to come up with an example. After tapping local knowledge it seems that I was right. There's a recent paper of Alexander Lubotzky that proves that if $G$ is a finite simple group then the only restriction on a subset $A\subseteq G$ for it to be the image of the word map for some word in 2 variables is that it contains the identity and is fixed by $\operatorname{Aut}(G)$. Since there are finite simple groups (e.g., the Mathieu group $M_{11}$) with elements that are not sent to their inverses by any automorphism, this answers the question.

However, my real question is whether there's a relatively simple example?

Lubotzky's paper doesn't give an explicit word, although it does show that, for $M_{11}$, there's a word that works with length at most about $1.7\times 10^{244552995}$.

Presumably one can do a bit better than that?

There are obvious restrictions on $g\in G$ and $w$ that rule out *really* small examples. There can't be any automorphism of $G$ sending $g$ to $g^{-1}$ or any automorphism of the free group $F_n$ sending $w$ to $w^{-1}$.