Let $A$ be a finite abelian group and $G$ a finite non-abelian groups and let $\alpha$, $\beta:G\rightarrow{\rm Aut}(A)$ two homomorphisms with the same non-trivical kernel $H$.
Suppose that $\alpha (G)$ and $\beta (G)$ are conjugate subgroups of ${\rm Aut}(A)$. Are the semidirect products $A\rtimes _{\alpha }G$ and $A\rtimes _{\beta }G$ isomorphic?
I know that this is not true in general, but I can't find a counterexample.
Thank you in advance.
Best Answer
There are examples with $A = C_3 \times C_5\ (\cong C_{15})$ and $G = D_8$ (dihedral of order $8$).
We have $A = \langle x,y \mid x^3=y^5=1,xy=yx \rangle$ and $G = \langle a,b \mid a^4=1, b^2=1, b^{-1}ab=a^{-1} \rangle$.
We define $\alpha$ and $\beta$ by $$ \alpha(a)(x)=x, \alpha(a)(y)=y^{-1}, \alpha(b)(x)=x^{-1}, \alpha(b)(y)=y,$$ $$ \beta(a)(x)=x^{-1}, \beta(a)(y)=y, \beta(b)(x)=x, \beta(b)(y)=y^{-1}.$$
Then $H:=\ker(\alpha) = \ker(\beta) = \langle a^2 \rangle$ and $G/H \cong C_2 \times C_2$ is abelian. It is clear that $\alpha(G) = \beta(G)$.
The two semidirect products are not isomorphic - they have different numbers of elements of order $6$ for example. In the GAP and Magma libraries they are $\mathtt{SmallGroup}(120,13)$ and $\mathtt{SmallGroup}(120,12)$.