Identify subgroup of a semidirect product in GAP

Question

I have created an external semidirect product of two groups $H$ and $K$ in GAP. Now, I want to identify $H$ as a subgroup of this semidirect product, but I am unable to identify $H$. How can I identify $H$ in this semidirect product in GAP? Please help!!! Thank you. Below is the code that I used to construct the semidirect product.

gap> H:=DihedralGroup(12);
<pc group of size 12 with 3 generators>
gap> aut:=AutomorphismGroup(H);
<group of size 12 with 3 generators>
gap> inn:=InnerAutomorphismsAutomorphismGroup(aut);
<group with 3 generators>
gap> semi:=SemidirectProduct(inn,H);
<pc group with 5 generators>

Now, I want to identify inn (InnerAutomorphism group of H) as the subgroup of this semidirect product.

Answer 1

The group constructed with SemidirectProduct will have two monomorphisms from the constituents. Embedding(semi,1) is the one from inn, Embedding(semi,2) the one from H. Thus its Image will be the group you want, and you can also use it to find the elements corresponding to particular inner automorphisms:

gap> ea:=Embedding(semi,1);;
gap> Image(ea);
Group([ f1, f2, f2^2 ])
gap> Image(ea,InnerAutomorphism(H,H.1));
f1