If F is a factor structure of G, with kernel N, complete information about the connection between F and G is provided by the natural homomorphism.
In GAP 3, the ``official way'' to construct this natural homomorphism
was to create first the factor structure F, and then to call
NaturalHomomorphism with the arguments G and F.
For that, the data necessary to compute the homomorphism was stored in
F when F was constructed.
In GAP 4, factor structures are not treated in a special way,
in particular they do not store information about a homomorphism.
Instead, the more natural way is taken to construct the natural
homomorphism from G and N by NaturalHomomorphismByNormalSubgroup
if N is a normal subgroup of the group G,
or by NaturalHomomorphismByIdeal if N is an ideal in the ring G.
The factor F can then be accessed as the image of this homomorphism,
and of course G is the preimage and N is the kernel.
Note that GAP 4 does not guarantee anything about the representation of the factor F, it may be a permutation group or a polycyclically presented group or another kind of group. Also note that a natural homomorphism need not be surjective.
A consequence of this change is that GAP 4 does not allow you to construct a natural homomorphism from the groups G and F.
The other common type of homomorphism in GAP 3, ``operation homomorphisms'', have been replaced (just a name change) by action homomorphisms, which are handled in a similar fashion. That is, an action homomorphism is constructed from an acting group, an action domain, and a function describing the operation. The permutation group arising by the induced action is then the image of this operation homomorphism.
The GAP 3 function Operation is still supported, under the name Action,
but from the original group and the result of Action it is not
possible to construct the action homomorphism.
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GAP 4 manual