  Let $\Gamma$ be a finite, undirected, connected, simple graph. We say that a
matching $\mathcal{M}$ is a \textit{permutable $m$-matching} if $\mathcal{M}$
contains $m$ edges and the subgroup of $\text{Aut}(\Gamma)$ that fixes the
matching $\mathcal{M}$ setwise allows the edges of $\mathcal{M}$ to be permuted
in any fashion. A matching $\mathcal{M}$ is \textit{2-transitive} if the
setwise stabilizer of $\mathcal{M}$ in $\text{Aut}(\Gamma)$ can map any ordered
pair of distinct edges of $\mathcal{M}$ to any other ordered pair of distinct
edges of $\mathcal{M}$. We provide constructions of graphs with a permutable
matching; we show that, if $\Gamma$ is an arc-transitive graph that contains a
permutable $m$-matching for $m \ge 4$, then the degree of $\Gamma$ is at least
$m$; and, when $m$ is sufficiently large, we characterize the locally
primitive, arc-transitive graphs of degree $m$ that contain a permutable
$m$-matching. Finally, we classify the graphs that have a $2$-transitive
perfect matching and also classify graphs that have a permutable perfect
matching.
