TITLE: Deciding if $\mathbb{Z}\ltimes_A \mathbb{Z}^5$ and $\mathbb{Z}\ltimes_B \mathbb{Z}^5$ are isomorphic or not
QUESTION [14 upvotes]: I asked this in this MSE question but I didn't get answers. I think maybe here someone can help me.
I have the two following groups
$G_A=\mathbb{Z}\ltimes_A \mathbb{Z}^5$, where $A=\begin{pmatrix} 1&0&0&1&0\\0&-1&0&0&0\\0&0&-1&0&0\\0&0&0&0&-1\\0&0&0&1&0\end{pmatrix}$ and
$G_B=\mathbb{Z}\ltimes_B \mathbb{Z}^5$, where $B=\begin{pmatrix} 1&0&0&1&0\\0&-1&0&1&0\\0&0&-1&0&0\\0&0&0&0&-1\\0&0&0&1&0\end{pmatrix}$.
The product is given (for example in $G_A$) by $(k,m)\cdot(\ell,n)=(k+\ell, m+A^k n)$.

Problem: Decide if $G_A$ is isomorphic to $G_B$ or not.

My thoughts: I think strongly that they are not isomorphic but I couldn't prove it. The matrices $A$ and $B$ are both of order 4, they're not conjugate in $\mathsf{GL}(n,\mathbb{Z})$ (neither $B$ and $A^{-1}$) but they are conjugate in $\mathsf{GL}(n,\mathbb{Q})$. In some other cases, I've seen that they're not isomorphic by computing the abelianization, but in this case both have the same abelianization, namely $\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}_2\oplus\mathbb{Z}_2$. Even worse, both have 1 as an eigenvalue.
In my previous MO question there is a counterexample for the implication "$G_A\cong G_B\Rightarrow A\sim B^{\pm 1}$" so I cannot use that.
Thanks!

REPLY [6 votes]: Here is Derek Holt's computation done in GAP:
gap> LoadPackage("anupq");
gap> F := FreeGroup("a","b","c","d","e","t");;
gap> AssignGeneratorVariables(F);
gap> comms := List(Combinations(GeneratorsOfGroup(F){[1..5]},2),Comm);;
gap> G1 := F/Concatenation(comms,
>                          [Comm(a,t),b^t*b,c^t*c,d^t*a^-1*e^-1,     e^t*d]);;
gap> G2 := F/Concatenation(comms,
>                          [Comm(a,t),b^t*b,c^t*c,d^t*b^-1*a^-1*e^-1,e^t*d]);;
gap> Pq(G1:Prime:=2,ClassBound:=2);
<pc group of size 512 with 9 generators>
gap> StructureDescription(last);
"(C4 x C4 x C4 x C2) : C4"
gap> Pq(G2:Prime:=2,ClassBound:=2);
<pc group of size 256 with 8 generators>
gap> StructureDescription(last);   
"C2 x ((C4 x C4 x C2) : C4)"