Theorem
Let $G,H$ be abelian groups such that $Dih(G)\cong Dih(H)$
If $G$ is finitely generated, then $G\cong H$.
I'm curious whether "finitely generated" hypothesis can be removed.
If it's not true in general, what would be a counterexample?
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EDIT:
The below statement is equivalent to the above question.
Let $G,H$ be abelian groups such that every element of $G,H$ is of order 2.
If $G\times Z_2 \cong H\times Z_2$ (direct product), then $G\cong H$.
Is this statement true?
For your second question, there is no need to assume that $G,H$ are abelian, because all groups in which every element has order $2$ are abelian! Such groups are vector spaces over the field of order $2$, so the groups $G$ and $H$ in your question are subspaces of $G \times C_2$, and they clearly have the same dimension, so they are isomorphic.