  In this article, we provide counterexamples to a conjecture of M. Pellegrini
and P. Shumyatsky which states that each coset of the centralizer of an
involution in a finite non-abelian simple group $G$ contains an odd order
element, unless $G=\text{PSL}(n,2)$ for $n\geq 4$. More precisely, we show that
the conjecture does not hold for the alternating group $A_{8n}$ for all $n\geq
2$.
