  By a 2-group we mean a groupoid equipped with a weakened group structure. It
is called split when it is equivalent to the semidirect product of a discrete
2-group and a one-object 2-group. By a permutation 2-group we mean the 2-group
$\mathbb{S}ym(\mathcal{G})$ of self-equivalences of a groupoid $\mathcal{G}$
and natural isomorphisms between them, with the product given by composition of
self-equivalences. These generalize the symmetric groups $\mathsf{S}_n$, $n\geq
1$, obtained when $\mathcal{G}$ is a finite discrete groupoid.
  After introducing the wreath 2-product $\mathsf{S}_n\wr\wr\ \mathbb{G}$ of
the symmetric group $\mathsf{S}_n$ with an arbitrary 2-group $\mathbb{G}$, it
is shown that for any (finite type) groupoid $\mathcal{G}$ the permutation
2-group $\mathbb{S}ym(\mathcal{G})$ is equivalent to a product of wreath
2-products of the form $\mathsf{S}_n\wr\wr\
\mathbb{S}ym(\mathcal{B}\mathsf{G})$, where $\mathcal{B}\mathsf{G}$ is the
delooping of $\mathsf{G}$. This is next used to compute the homotopy invariants
of $\mathbb{S}ym(\mathcal{G})$ which classify it up to equivalence. In
particular, we prove that $\mathbb{S}ym(\mathcal{G})$ can be non-split, and
that the step from the trivial groupoid $\mathcal{B}\mathsf{1}$ to an arbitrary
one-object groupoid $\mathcal{B}\mathsf{G}$ is in fact the only source of
non-splitness. Various examples of permutation 2-groups are explicitly
computed, in particular the permutation 2-group of the underlying groupoid of a
(finite type) 2-group. It also follows from well known results about the
symmetric groups that the permutation 2-group of the groupoid of all finite
sets and bijections between them is equivalent to the direct product 2-group
$\mathbb{Z}_2[1]\times\mathbb{Z}_2[0]$, where $\mathbb{Z}_2[0]$ and
$\mathbb{Z}_2[1]$ stand for the group $\mathbb{Z}_2$ thought of as a discrete
and a one-object 2-group, respectively.
