  Given a graph $X$ with a Hamilton cycle $C$, the {\em compression factor
$\kappa(X,C)$ of $C$} is the order of the largest cyclic subgroup of
$\operatorname{Aut}(C)\cap\operatorname{Aut}(X)$, and the {\em Hamilton
compression $\kappa(X)$ of $X$ } is the maximum of $\kappa(X,C)$ where $C$ runs
over all Hamilton cycles in $X$. Generalizing the well-known open problem
regarding the existence of vertex-transitive graphs without Hamilton
paths/cycles, it was asked by Gregor, Merino and M\"utze in [``The Hamilton
compression of highly symmetric graphs'', {\em arXiv preprint} arXiv:
2205.08126v1 (2022)] whether for every positive integer $k$ there exists
infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton
compression equal to $k$. Since an infinite family of Cayley graphs with
Hamilton compression equal to $1$ was given there, the question is completely
resolved in this paper in the case of Cayley graphs with a construction of
Cayley graphs of semidirect products $\mathbb{Z}_p\rtimes\mathbb{Z}_k$ where
$p$ is a prime and $k \geq 2$ a divisor of $p-1$. Further, infinite families of
non-Cayley vertex-transitive graphs with Hamilton compression equal to $1$ are
given. All of these graphs being metacirculants, some additional results on
Hamilton compression of metacirculants of specific orders are also given.
