Let $G$ and $H$ be two groups and $\alpha:G\rightarrow H, \beta:H\rightarrow G$ be two maps between them.
Under what conditions can we say that $G\cong H$? Note that we do not require for $\alpha$ or $\beta$ to be an isomorphism. Just for there to exist one.
i) For example, is it true if both $\alpha$ and $\beta$ are surjective?
ii) It is clearly not true if they are injective. But assume they are both injective. What else do we need for $G$ and $H$ to be isomorphic?
I realise that this is a mathematically badly worded question, as it has obvious answers (such as if they are additionally surjective, haha). But I hope that the question can still be understood. More "categorical" properties are more desirable, but I am generally interested in all of them.
Thanks all in advance.
(Note, groups are not assumed to be abelian).
EDIT: Maybe this is a better way to formulate. What property can I demand on a morphism between groups, that is NOT an isomorphism, BUT if I have such morphisms on both sides, together, they become an isomorphism.
An example as I said, is surjectivity.