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By a 2-group we mean a groupoid equipped with a weakened group structure. Itis called split when it is equivalent to the semidirect product of a discrete2-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 ofself-equivalences. These generalize the symmetric groups $\mathsf{S}_n$, $n\geq1$, obtained when $\mathcal{G}$ is a finite discrete groupoid.  After introducing the wreath 2-product $\mathsf{S}_n\wr\wr\ \mathbb{G}$ ofthe symmetric group $\mathsf{S}_n$ with an arbitrary 2-group $\mathbb{G}$, itis shown that for any (finite type) groupoid $\mathcal{G}$ the permutation2-group $\mathbb{S}ym(\mathcal{G})$ is equivalent to a product of wreath2-products of the form $\mathsf{S}_n\wr\wr\\mathbb{S}ym(\mathcal{B}\mathsf{G})$, where $\mathcal{B}\mathsf{G}$ is thedelooping of $\mathsf{G}$. This is next used to compute the homotopy invariantsof $\mathbb{S}ym(\mathcal{G})$ which classify it up to equivalence. Inparticular, we prove that $\mathbb{S}ym(\mathcal{G})$ can be non-split, andthat the step from the trivial groupoid $\mathcal{B}\mathsf{1}$ to an arbitraryone-object groupoid $\mathcal{B}\mathsf{G}$ is in fact the only source ofnon-splitness. Various examples of permutation 2-groups are explicitlycomputed, in particular the permutation 2-group of the underlying groupoid of a(finite type) 2-group. It also follows from well known results about thesymmetric groups that the permutation 2-group of the groupoid of all finitesets 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 discreteand a one-object 2-group, respectively.

Permutation 2-groups I: structure and splitness