  Greither and Pareigis have established a connection between Hopf Galois
structures on a Galois extension $L/K$ with Galois group $G$, and the regular
subgroups of the group of permutations on $G$, which are normalized by $G$.
Byott has rephrased this connection in terms of certain equivalence classes of
injective morphisms of $G$ into the holomorph of the groups $N$ with the same
cardinality of $G$.
  Childs and Corradino have used this theory to construct such Hopf Galois
structures, starting from fixed-point-free endomorphisms of $G$ that have
abelian images. In this paper we show that a fixed-point-free endomorphism has
an abelian image if and only if there is another endomorphism that is its
inverse with respect to the circle operation in the near-ring of maps on $G$,
and give a fairly explicit recipe for constructing all such endomorphisms.
