  Let VB$_n$ be the virtual braid group on $n$ strands and let $\mathfrak{S}_n$
be the symmetric group on $n$ letters. Let $n,m \in \mathbb{N}$ such that $n
\ge 5$, $m \ge 2$ and $n \ge m$. We determine all possible homomorphisms from
VB$_n$ to $\mathfrak{S}_m$, from $\mathfrak{S}_n$ to VB$_m$ and from VB$_n$ to
VB$_m$. As corollaries we get that Out(VB$_n$) is isomorphic to
$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ and that VB$_n$ is both
Hopfian and co-Hofpian.
