Realizing an isomorphism of (faithful) semidirect products Suppose that we have two finite groups $A$ and $B$, and suppose that $A$ is abelian. Let $G_{\phi} = A \rtimes_{\phi} B$ be the semidirect product given by $\phi: B \to \operatorname{Aut}(A)$ , and let $G_{\psi} = A \rtimes_{\psi} B$ be another semidirect product given by $\psi: B \to \operatorname{Aut}(A)$. Suppose that both $\psi$ and $\phi$ are injective maps. 
I want to show that if $G_\phi \simeq G_\psi$ then $\phi(B)$ and $\psi(B)$ are in the same conjugacy class in $\operatorname{Aut}(A)$, or to find a counterexample. I suspect this is not true in general, but I am struggling to find a counterexample.
 A: For a counterexample, let $A = S_4$, let $|B |=2$ with $B = \langle b \rangle$, and suppose that $\phi(b)$ and $\psi(b)$ are respectively the inner automorphisms defined by conjugation by the elements $(1,2)$ and $(1,2)(3,4)$ of $S_4$.
Then $\phi(B)$ and $\psi(B)$ are not conjugate in ${\rm Aut}(A) = S_4$, but the fact that $\phi(b)$ and $\psi(b)$ are inner automorphisms implies that $G_\phi \cong G_\psi \cong A \times B$.
I realize now that the above example does not answer the question asked, because $A$ is not abelian.
I found an counterexample in which $A$ is abelian by computer search, which is not completely satisfactory perhaps, but it answers the question.
The groups $G_\phi$ and $G_\psi$ are isomorphic to the group $\mathtt{SmallGroup}(324,121)$ in the GAP/Magma small groups library.
This group has two distinct elementary abelian subgroups of order $27$ (which become $A$ in $G_\phi$ and $G_\psi$), and each has complements $B$ isomorphic to the dihedral group of order $12$. But these complements induce non-conjugate groups of automorphisms of $A$ in the two cases. (The element of order $2$ in $Z(B)$ centralizes subgroups of order $1$ and $3$ of $A$ in the two cases.)
Here is some Magma code that I used for finding this example. I had already tried with $A$ elementary abelian of order $4,8,9 without finding an example.
G := AGL(3,3); //semidriect product of C3^3 with its full automorphism group
Q,phi,P := pCoreQuotient(G,3);  //Quotient of G by C3^3
S := [s`subgroup @@ phi : s in Subgroups(Q)]; #S; //this quotient has 154 subgroups
for i in [1..#S-1] do for j in [i+1..#S] do
  if IsIsomorphic(S[i],S[j]) and IsIsomorphic(quo<S[i]|P>,quo<S[j]|P>) then
  i,j;
end if; end for; end for;
// 38 41 was printed, but this varies from run to run
IdentifyGroup(S[38]); //<324, 121>
IdentifyGroup(S[43]); //<324, 121>

