Isomorphism preserving operations

Questions : What are group modification processes ( group transformations ) which preserve the isomorphism property. One operation is abelianization of group does not preserve isomorphism. I mean to say if G and H are isomorphic iff G' and H' are also isomorphic. Where G' and H' are groups after applying group modification process.

• What counts as a "modification"? Groups are rarely determined by smaller groups. – Tobias Kildetoft Jun 14 '18 at 12:24
• Extending groups to isomorphic supergroups seems equally unlikely to be helpful because you want an equivalent condition. – N8tron Jun 14 '18 at 12:27
• Tietze transformations are a nice example of something related (but note quite related enough to make this comment an answer!). Group presentations are a way of viewing groups, and Tietze transformations are transformations applied to presentations (so they change the "view" of your group). Then two presentations $\mathcal{P}$ and $\mathcal{Q}$ define isomorphic groups if and only if $\mathcal{Q}$ can be obtained from $\mathcal{P}$ via a sequence of Tietze transformations. – user1729 Jun 14 '18 at 12:58