I am really new in GAP, please accept my apologize if my question is a simple one.
I want to implement an algorithm in GAP for the next problem:
Given an abelian finite group G, find the group H such as$$|Aut(Z \times G)| = |Aut(Z\times H)|$$ Honestly, I do not have a clear idea of how to implement it. I began with the calculation of direct product $\mathbb(Z \times G)$, using $DirectProduct(FreeGroup(1), G)$, then I called the following method, $AutomorphismGroup(Z\times G)$, in order to get the automorphism group of $Z \times G$. I am not sure if this is a good approach, but it is the only idea that came up at the moment.
Many thanks in advance!