Let
L
|
K
L|K
be a Galois extension of fields with finite Galois group
G
G
. Greither and Pareigis showed that there is a bijection between Hopf Galois structures on
L
|
K
L|K
and regular subgroups of
P
e
r
m
(
G
)
Perm(G)
normalized by
G
G
, and Byott translated the problem into that of finding equivalence classes of embeddings of
G
G
in the holomorph of groups
N
N
of the same cardinality as
G
G
. In 2007 we showed, using Byott’s translation, that fixed point free endomorphisms of
G
G
yield Hopf Galois structures on
L
|
K
L|K
. Here we show how abelian fixed point free endomorphisms yield Hopf Galois structures directly, using the Greither-Pareigis approach and, in some cases, also via the Byott translation. The Hopf Galois structures that arise are “twistings” of the Hopf Galois structure by
H
λ
H_{\lambda }
, the
K
K
-Hopf algebra that arises from the left regular representation of
G
G
in
P
e
r
m
(
G
)
Perm(G)
. The paper concludes with various old and new examples of abelian fixed point free endomorphisms.