  We determine all finite p-groups that admit a faithful, self-similar action
on the p-ary rooted tree such that the first level stabilizer is abelian. A
group is in this class if and only if it is a split extension of an elementary
abelian p-group by a cyclic group of order p.
  The proof is based on use of virtual endomorphisms. In this context the
result says that if G is a finite p-group with abelian subgroup H of index p,
then there exists a virtual endomorphism of G with trivial core and domain H if
and only if G is a split extension of H and H is an elementary abelian p-group.
