Finiteness of $z$-classes in reductive groups

Let $k$ be a perfect field such that for every $n$ there are only finitely many field extensions, up to isomorphism, of $k$ of degree $n$. If $G$ is a reductive algebraic group defined over $k$, whose characteristic is very good for $G$, then we prove that $G(k)$ has only finitely many $z$-classes. For each perfect field $k$ which does not have the above finiteness property we show that there exist groups $G$ over $k$ such that $G(k)$ has infinitely many $z$-classes.