Dear forum,
s4:=SymmetricGroup(4)
gap> t;
Group([ (1,4,3), (1,4) ]) is a subgroup of s4, but not normal
gap> v;
Group([ (1,2)(3,4), (1,3)(2,4) ]) is a normalsubgroup of s4
Because t is a complement to v in s4, s4 is a semidirectproduct of t by v.
But using the automorphismGroup of t all the
GroupHomomorphismByImages failed and the construction of the
semidirectProduct(s,v)
was impossinble.
Are there other methods?

A second question:
G/H=Q
knowing H and Q can GAP construct G or the isomorphismClass of G?

Many thanks and
best wishes
K.Ewald