  The existence of $1$-factorizations of an infinite complete equipartite graph
$K_m[n]$ (with $m$ parts of size $n$) admitting a vertex-regular automorphism
group $G$ is known only when $n=1$ and $m$ is countable (that is, for countable
complete graphs) and, in addition, $G$ is a finitely generated abelian group
$G$ of order $m$.
  In this paper, we show that a vertex-regular $1$-factorization of $K_m[n]$
under the group $G$ exists if and only if $G$ has a subgroup $H$ of order $n$
whose index in $G$ is $m$. Furthermore, we provide a sufficient condition for
an infinite Cayley graph to have a regular $1$-factorization. Finally, we
construct 1-factorizations that contain a given subfactorization, both having a
vertex-regular automorphism group.
