Quasi-inverse endomorphisms

Greither and Pareigis have established a connection between Hopf Galois structures on a Galois extension $L\mathrm{/}K$ 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.