A note on groups in which the centralizer of every element of order $5$ is a $5$-group

The main theorem in this article shows that a group of odd order which admits the alternating group of degree $5$ with an element of order $5$ acting fixed point freely is nilpotent of class at most $2$. For all odd primes $r$, other than $5$, we give a class $2$ $r$-group which admits the alternating group of degree $5$ in such a way. This theorem corrects an earlier result which asserts that such class $2$ groups do not exist. The result allows us to state a theorem giving precise information about groups in which the centralizer of every element of order $5$ is a $5$-group.